Title: Theory / Important / Math / Analysis / Zorich{ / 01 Limits and continuity; / 1 Single Variable;
Tags: fromMath mathToDef mathToSee examplesSwitchExample1 relation_1582033777334 fromMathToDo hasSeenConjDef EquivalentButNoIntermediate TypToQuestion addIntermediateCard closed exampleToQuestion multipartite removableComplex récupéré typToBinary

Name: A sequence
Definition: A function whose domain is \(\mathbb N\).
Page: 79
Chapter: 3
Section: 1
Kind: Def

Name: The limit of \(\{x_i\}\in\mathbb R\)
Definition: A real \(A\) such that for every neighborhood \(V(A)\) of \(A\) there exists an index \(N\) (depending on \(V(A)\)) such that all terms of the sequence having index larger than \(N\) belongs to the neighborhood of \(V(A)\).
Page: 80
Chapter: 3
Section: 1
Variables: \(\{x_i\}\in\mathbb R\)
Index: 2
Kind: Def

Name: A divergent sequence
Definition: A sequence which does not converge.
Page: 80
Chapter: 3
Section: 1
Kind: Def

Question: \(\lim_{n\to\infty}\frac 1n\)
Answer: is 0.
Page: 80
Chapter: 3
Section: 1
Kind: Ex

Question: \(\lim_{n\to\infty}\frac {n+1}{n}\)
Answer: is 1.
Page: 81
Chapter: 3
Section: 1
Index: 2
Kind: Ex

Question: \(\lim_{n\to\infty}1+\frac {(-1)^n}{n}\)
Answer: is 1.
Page: 81
Chapter: 3
Section: 1
Kind: Ex

Question: \(\lim_{n\to\infty}\frac{\sin n}{n}\)
Answer: is 0.
Page: 81
Chapter: 3
Section: 1
Index: 4
Kind: Ex

Question: \(\lim_{n\to\infty}\frac{1}{q^n}\)
Answer: is 0.
Page: 81
Chapter: 3
Section: 1
Variables: \(q\in\mathbb R\) with [$]\abs{q}>1[/$]
Index: 5
Kind: Ex

Question: \(\lim_{n\to\infty}n^{(-1)^n}\)
Answer: diverges.
Page: 81
Chapter: 3
Section: 1
Variables: \(q\in\mathbb R\) with [$]\abs{q}>1[/$]
Index: 6
Kind: Ex

Question: \(\lim_{n\to\infty}(-1)^n\)
Answer: diverges.
Page: 81
Chapter: 3
Section: 1
Index: 7
Kind: Ex

Name: Ultimately constant sequence
Definition: A sequence \(\{x_i\}\) such that there exists \(N\in\mathbb N\), \(x_i=x_j\) for all \(i,j>N\).
Page: 81
Chapter: 3
Section: 1.2.a
Index: 4
Kind: Def

Name: Bounded sequence
Definition: A sequence \(\{x_i\}\) such that there exists \(M>0\), such that [$]\abs{x_i}<M[/$] for all \(i\in\mathbb N\).
Page: 81
Chapter: 3
Section: 1.2.a
Index: 5
Kind: Def

Smaller0: \(\{x_i\}\) ultimately constant
Greater0: \(\{x_i\}\) converges
Page: 82
Chapter: 3
Section: 1.2.a
Variables: \(\{x_i\}\) a sequence
Index: 1 a)
Kind: Theo

Smaller0: \(\{x_i\}\) converges
Greater0: \(\{x_i\}\) is bounded
Page: 82
Chapter: 3
Section: 1.2.a
Variables: \(\{x_i\}\) a sequence,
\(A\in\mathbb R\)
Index: 1 d)
Kind: Theo

Question: The number of limits is
Answer: at most \(1\).
Page: 82
Chapter: 3
Section: 1.2.a
Variables: \(\{x_i\}\) a sequence
Index: 1 c)
Kind: Theo

Name: The product of \(\{x_n\}\) and \(\{y_n\}\)
Definition: \(\{(x_n\times y_n)\}\), \(\{(x_n/y_n)\}\) if \(y_n\ne0\).
Page: 82
Chapter: 3
Section: 1.2.b
Variables: \(\{x_n\}\), \(\{y_n\}\) sequences
Notation: \(\{x_n\}\times\{y_n\}\)
Index: 6
Kind: Def

Name: The sum of \(\{x_n\}\) and \(\{y_n\}\)
Definition: \(\{(x_n+y_n)\}\)
Page: 82
Chapter: 3
Section: 1.2.b
Variables: \(\{x_n\}\), \(\{y_n\}\) sequences
Notation: \(\{x_n\}+\{y_n\}\)
Index: 6
Kind: Def

Name: The quotient of \(\{x_n\}\) and \(\{y_n\}\)
Definition: \(\{(x_n/y_n)\}\) if \(y_n\ne0\).
Page: 82
Chapter: 3
Section: 1.2.b
Variables: \(\{x_n\}\), \(\{y_n\}\) sequences
Notation: \(\{x_n\}/\{y_n\}\)
Index: 6
Kind: Def

Smaller0: \(X<Y\)
Greater0: there exists \(N\in\mathbb N\) such that \(x_i<y_i\) for each \(i>N\)
Page: 82
Chapter: 3
Section: 1.2.b
Variables: \(\{x_n\}\) sequence with limit \(X\),
\(\{y_n\}\) sequence with limit \(Y\)
Kind: Theo

Smaller0: \(A=C\)
Greater0: \(B=C\)
Page: 82
Chapter: 3
Section: 1.2.b
Variables: \(\{x_n\}\), \(\{y_n\}\) and \(\{z_n\}\) sequences with limit \(A\), \(B\) and \(C\)
Kind: Theo
Smaller1: \(x_i\le y_i\le z_i\) for \(i\) great enough

Smaller0: \(x_n>y_n\), \(x_n\ge y_n\), \(x_n>B\) or \(x_n\ge B\) for \(n\) great enough
Greater0: \(A\ge B\)
Page: 82
Chapter: 3
Section: 1.2.b
Variables: \(\{x_n\}\), \(\{y_n\}\) sequences with limit \(A\), \(B\).
Kind: Cor

Name: A Cauchy sequence
Definition: A sequence \(\{x_n\}\) such that [$]d(x_m,x_n)=\abs{x_m-x_n}<\epsilon[/$] for \(m,n\) great enough.
Page: 82, 417
Chapter: 3, 7
Section: 1.3.a, 2.1
Index: 7, 4
Kind: Def, Def
Name2: A fundamental sequence

Question: The sequence \(\{0.\alpha_1\dots\alpha_n\}_{n\in\mathbb N}\)
Answer: converges.
Page: 87
Chapter: 3
Section: 1.3.a
Variables: \(\{\alpha_n\}\in\{0,1\}\)
Index: 9
Kind: Ex

Question: \(\lim_{n\to+\infty}\{\sum_{i=1}^n\frac 1i\}\)
Answer: equals \(+\infty\) (diverges)
Page: 87, 96
Chapter: 3, 3
Section: 1.3.a, 1.4.a
Index: 10, 21
Kind: Ex

Name: Increasing/decreasing/nonincreasing/nondecreasing sequence
Definition: A sequence \(\{x_n\}\) such that \(x_i<x_{i+1}\)/\(x_i>x_{i+1}\)/\(x_i\ge x_{i+1}\)/\(x_i\le x_{i+1}\).
Page: 87
Chapter: 3
Section: 1.3.b
Index: 8
Kind: Def

Name: Bounded above sequence
Definition: A sequence \(\{x_n\}\) such that \(x_n<M\) for some \(M\in\mathbb R\).
Page: 87
Chapter: 3
Section: 1.3.b
Index: 9
Kind: Def

Smaller0: \(\{x_n\}\) is nondecreasing
Greater0: \(\{x_n\}\) is bounded by above
Page: 88
Chapter: 3
Section: 1.3.b
Variables: \(\{x_n\}\) a sequence
Index: 5
Kind: Theo
Name: Weierstrass Theorem
Greater1: \(\{x_n\}\) is converging
Connect Greater0: iff

Question: \(\lim _{n\to\infty}\frac n{q^n}\)
Answer: is \(0\).
Page: 88
Chapter: 3
Section: 1.3.b
Variables: \(q>1\).
Index: 11
Kind: Ex

Question: \(\lim _{n\to\infty}\sqrt[n]{n}\)
Answer: is \(1\).
Page: 88
Chapter: 3
Section: 1.3.b
Index: 1
Kind: Cor

Question: \(\lim _{n\to\infty}\sqrt[n]{a}\)
Answer: is \(1\).
Page: 89
Chapter: 3
Section: 1.3.b
Variables: \(a>0\)
Index: 2
Kind: Cor

Question: \(\lim _{n\to\infty}\frac{q^n}{n!}\)
Answer: is \(0\).
Page: 89
Chapter: 3
Section: 1.3.b
Variables: \(q\in\mathbb R\)
Index: 12
Kind: Ex

Name: A subsequence of \(\{x_i\}\)
Definition: \(\{x_{n_i} \}\) for \(\{n_i\}\) an increasing sequence of natural numbers.
Page: 89
Chapter: 3
Section: 1.3.d
Variables: \(\{x_i\}\) a sequence
Index: 10
Kind: Def

Name: \(\{x_n\}\) tends to infinity
Definition: For each \(c\in\mathbb R\), [$]\abs{x_i}>c[/$] for \(i\) great enough.
Page: 91
Chapter: 3
Section: 1.3.d
Variables: \(\{x_N\}\in\mathbb R\) a sequence
Notation: [$]\abs{x_n}\to+\infty[/$]
Index: 12
Kind: Def

Name: The inferior limit of \(\{x_n\}\)
Definition: \(\lim_{n\to\infty}\text{inf}_{k\ge n}x_k\).
Page: 91
Chapter: 3
Section: 1.3.d
Variables: \(\{x_n\}\in\mathbb R\) a sequence
Notation: \(\underline{\lim}_{k\to\infty}x_k\)
Index: 13
Kind: Def

Name: The superior limit of \(\{x_n\}\)
Definition: \(\lim_{n\to\infty}\text{sup}_{k\ge n}x_k\).
Page: 91
Chapter: 3
Section: 1.3.d
Variables: \(\{x_n\}\in\mathbb R\) a sequence
Notation: \(\overline{\lim}_{k\to\infty}x_k\)
Index: 13
Kind: Def

Question: \(\underline\lim_{n\to\infty} (-1)^n\) is
Answer: -1
Page: 92
Chapter: 3
Section: 1.3.d
Index: 14
Kind: Ex

Question: \(\underline\lim_{n\to\infty} n^{(-1)^n}\) is
Answer: 0
Page: 92
Chapter: 3
Section: 1.3.d
Index: 15
Kind: Ex

Question: \(\underline\lim_{n\to\infty} n\) is
Answer: \(+\infty\)
Page: 92
Chapter: 3
Section: 1.3.d
Index: 16
Kind: Ex

Question: \(\underline\lim_{n\to\infty} \frac{(-1)^k}{k}\) is
Answer: 0.
Page: 92
Chapter: 3
Section: 1.3.d
Index: 17
Kind: Ex

Question: \(\underline\lim_{n\to\infty} -n^2\) is
Answer: \(-\infty\).
Page: 92
Chapter: 3
Section: 1.3.d
Index: 18
Kind: Ex

Question: \(\underline\lim_{n\to\infty} (-1)^nn\) is
Answer: \(-\infty\).
Page: 92
Chapter: 3
Section: 1.3.d
Index: 19
Kind: Ex

Question: \(\overline\lim_{n\to\infty} (-1)^nn\) is
Answer: \(+\infty\).
Page: 92
Chapter: 3
Section: 1.3.d
Index: 19
Kind: Ex

Question: \(\overline\lim_{n\to\infty} -n^2\) is
Answer: \(-\infty\).
Page: 93
Chapter: 3
Section: 1.3.d
Index: 18
Kind: Ex

Question: \(\overline\lim_{n\to\infty} \frac{(-1)^n}{n}\) is
Answer: \(0\).
Page: 92
Chapter: 3
Section: 1.3.d
Index: 17
Kind: Ex

Question: \(\overline\lim_{n\to\infty}n\) is
Answer: \(+\infty\).
Page: 92
Chapter: 3
Section: 1.3.d
Index: 16
Kind: Ex

Question: \(\overline\lim_{n\to\infty}n^{(-1)^n}\) is
Answer: \(+\infty\).
Page: 92
Chapter: 3
Section: 1.3.d
Index: 15
Kind: Ex

Question: \(\overline\lim_{n\to\infty}{(-1)^k}\) is
Answer: \(1\).
Page: 92
Chapter: 3
Section: 1.3.d
Index: 14
Kind: Ex

Name: A partial limit of \(\{x_n\}\)
Definition: The limit of a subsequence of \(\{x_n\}\).
Page: 93
Chapter: 3
Section: 1.3.d
Variables: \(\{x_n\}\in\mathbb R\)
Index: 15
Kind: Def

Name: The inferior and superior limits of \(\{x_n\}\)
Definition: the smallest and largest partial limits of \(\{x_n\}\)
Page: 93, 94
Chapter: 3
Section: 1.3.d
Variables: \(\{x_n\}\in\mathbb R\)
Index: 1, 1'
Kind: Prop
Extra: 1 is on bounded sequence, not 1'

Smaller0: \(\{x_n\}\) has limits (including \(\pm\infty\))
Greater0: its inferior and superior limits are the same
Page: 94
Chapter: 3
Section: 1.3.d
Variables: \(\{x_n\}\in\mathbb R\)
Index: 3, 4
Kind: Cor
Smaller to greater: iff
Intermediate: every subsequence converges

Definition: \(a_p+\dots+a_q\)
Page: 95
Chapter: 3
Section: 1.4 a
Notation: \(\sum_{n=p}^qa_n\)

Definition: \(a_p+\dots\)
Page: 95
Chapter: 3
Section: 1.4 a
Notation: \(\sum_{n=p}^{+\infty}a_n\)
Index: 16
Kind: Def

Name: A term of the series
Definition: Some \(a_i\)
Page: 95
Chapter: 3
Section: 1.4 a
Variables: \(\sum_{n=1}^\infty a_n\) a series.
Index: 17
Kind: Def

Name: The \(n\)-th partial sum of the series
Definition: \(\sum_{i=1}^na_i\)
Page: 95
Chapter: 3
Section: 1.4 a
Variables: \(\sum_{i=1}^\infty a_i\) a series
\(n\in\mathbb N\)
Notation: \(s_n\)
Index: 18
Kind: Def

Name: A convergent series
Definition: A series \(\sum a_i\) such that \(s_i\) converges.
Page: 95
Chapter: 3
Section: 1.4 a
Index: 19
Kind: Def

Name: The sum of the series
Definition: The limit of \(s_n\) if it exists.
Page: 95
Chapter: 3
Section: 1.4 a
Variables: \(\sum a_i\) a series
Index: 20
Kind: Def

Name: Cauchy convergence criterion for a series
Definition: For each \(\epsilon>0\), there exists \(N>0\) such that \(\sum_{n=p}^qa_n<\epsilon\) for \(p,q>N\).
Page: 95
Chapter: 3
Section: 1.4 a
Variables: \(\sum a_i\) a series
Index: 6
Kind: Theo

Smaller0: both series are identic appart from a finite number of terms
Greater0: both series converges or they both diverges
Page: 95
Chapter: 3
Section: 1.4 a
Variables: \(\sum a_i\), \(\sum b_i\) series
Index: 6
Kind: Cor
Hide middle: yes

Smaller0: the series converges
Greater0: \(\lim a_i=0\)
Page: 95
Chapter: 3
Section: 1.4 a
Variables: \(\sum a_i\)
Index: 7
Kind: Cor

Name: A geometric series
Definition: \(\sum q^n\) for some number \(q\).
Page: 96
Chapter: 3
Section: 1.4 a
Index: 20
Kind: Ex

Question: \(\sum q^n\)
Answer: converges iff [$]\abs{q}<1[/$]
Page: 96
Chapter: 3
Section: 1.4 a
Variables: \(q\in\mathbb R\)
Index: 20
Kind: Ex

Name: The harmonic series
Definition: \(\sum \frac1n\).
Page: 96
Chapter: 3
Section: 1.4 a
Index: 21
Kind: Ex

Question: \(\sum (-1)^n\)
Answer: diverges.
Page: 96
Chapter: 3
Section: 1.4 a
Index: 22
Kind: Ex

Name: Absolutely convergent series
Definition: [$]\sum_{n=1}^\infty\abs{a_n}[/$] converges.
Page: 96, 270
Chapter: 3, 5
Section: 1.4 b, 5.2
Index: 21, 4
Kind: Def, prop
Definition2: every rearangement of the series converges to the same sum
Conjdef: A series \(\sum_{i=1}^\infty a_i\), such that, equivalently

Smaller0: \(\sum a_i\) is absolutely convergent
Greater0: \(\sum a_i\) converges
Page: 96
Chapter: 3
Section: 1.4 b
Variables: \(\sum a_i\) a series
Index: 21
Kind: Def

Name: Convergent, non absolutely convergent, series
Page: 97
Chapter: 3
Section: 1.4 b
Variables: \(\sum a_i\) a series
Index: 23
Kind: Ex
Example: \(\sum(-1)^{i}\frac1{\lfloor \frac i2\rfloor}\)

Smaller0: the series converges
Greater0: the series is bounded above
Page: 98
Chapter: 3
Section: 1.4 b
Variables: \(\sum a_i\) a series
Index: 7
Kind: Theo
Smaller to greater: iff
Assuming: \(a_i\ge0\)

Smaller0: \(a_i,b_i\ge0\)
Greater0: \(a_i\) converges
Page: 98
Chapter: 3
Section: 1.4 b
Variables: \(\sum a_i\), \(\sum b_i\) series
Index: 8
Kind: Theo
Smaller1: \(a_i\le b_i\) for \(i \) great enough
Name: Comparison Theorem
Connect Smaller0: and
Smaller2: \(\sum b_i\) converges

Smaller0: [$]\abs{a_i}\le b_i[/$] for \(i\) great enough
Greater0: \(a_i\) converges absolutely
Page: 98
Chapter: 3
Section: 1.4 b
Variables: \(\sum a_i\), \(\sum b_i\) series
Index: 8
Kind: Cor
Smaller1: \(\sum b_i\) converges
Name: Weierstrass \(M\)-test
Extra: doublon nom
Connect Smaller0: and

Smaller0: [$]\overline{\lim}_{n\to \infty}\sqrt[n]{\abs{a_n}}[/$] is smaller than 1
Greater0: \(\sum a_i\) converges absolutely
Page: 98
Chapter: 3
Section: 1.4 b
Variables: \(\sum a_i\) series
Index: 9
Kind: Cor
Name: Cauchy test

Smaller0: [$]\overline{\lim}_{n\to \infty}\sqrt[n]{\abs{a_n}}[/$] is greater than 1
Greater0: \(\sum a_i\) diverges
Page: 98
Chapter: 3
Section: 1.4 b
Variables: \(\sum a_i\) series
Index: 9
Kind: Cor
Name: Cauchy test

Smaller0: \(\{a_i\}\) decreases and is positive
Greater0: \(\sum a_i\) converges
Page: 100
Chapter: 3
Section: 1.4 b
Variables: \(\sum a_i\) series
Index: 2
Kind: Prop
Hide middle: yes
Smaller1: \(\sum2^ka_{2^k}\) converges
Connect Smaller0: and

Question: Does \(\sum_{n\in\mathbb N}\frac{1}{n^p}\) converges ?
Answer: iff \(p>1\).
Page: 101
Chapter: 3
Section: 1.4 b
Variables: \(p\in\mathbb R\)
Kind: Cor

Name: Euler number
Definition: \(\sum\frac{1}{n!}\)
Page: 102, 89 https://en.wikipedia.org/wiki/E_(mathematical_constant)#Alternative_characterizations
https://en.wikipedia.org/wiki/List_of_limits
Chapter: 3, C
Section: 1.4 c, 1.3 c
Notation: \(e\)
Index: 13
Kind: Cor, Ex
Definition2: \(\lim _{n\to\infty}\left(1+\frac{1}{n}\right)^n\)
Definition3: \(\ln^{-1}(1)\)
Extra: That's not Euler(-Mascheroni) constant
Definition4: continued fraction
\([ 2; (1, 2n, 1,)_{n\in\mathbb N^*}, \dots]=[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, \dots, 1, 2n, 1, \dots]\),
 i.e. [$]2 +
\cfrac{1}
   {1 + \cfrac{1}
      {2 + \cfrac{1}
         {1 + \cfrac{1}
            {1 + \cfrac{1}
               {4 + \cfrac{1}
            {1 + \cfrac{1}
               {1 + \ddots}
                  }
               }
            }
         }
      }
   }
.[/$]
Definition5: continued fraction \([1;(4n+2,)_{n\in\mathbb N}]=[1; 2,6,10, 14, 18, 22, 26,\dots]\) i.e.
\(1 + \cfrac{2}{1 + \cfrac{1}{6 + \cfrac{1}{10 + \cfrac{1}{14 + \cfrac{1}{18 + \cfrac{1}{22 + \cfrac{1}{26 \ddots}}}}}}}\).
Definition6: \(\mathbf E(\min\{n\mid \sum_{i=1}^nX_i{\gt}1\})\) with each \(X_i\) uniform on \([0,1]\) and independant
Definition8: \(\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}}\)

Name: A limit point of \(f\) as \(x\in E\) tends to \(a\)
Definition: A real \(A\) such that, for every \(\epsilon>0\), there exists \(\delta>0\) such that [$]\abs{f(x)-A}<\epsilon[/$] for every \(x\in E\) such that [$]0<\abs{x-a}<\delta[/$].
Chapter: 3
Section: 2.1
Variables: \(E\subseteq \mathbb R\), \(a\) a limit point of \(E\) and \(f:E\to\mathbb R\)
Notation: \[\lim_{E\ni x\to a}f(x)\]
Index: 1
Kind: Def

Question: \[\lim_{\mathbb R\setminus\{0\}\ni x\to 0}x\sin{\frac1x}=\]
Answer: \[0\]
Page: 107
Chapter: 3
Section: 2.1
Index: 1
Kind: Ex

Name: A deleted neighborhood of \(a\)
Definition: A neighborhood of \(a\), minus \(\{a\}\).
Page: 107
Chapter: 3
Section: 2.1
Variables: \(a\in \mathbb R\)
Index: 2
Kind: Def

Definition: \(U(a)\setminus\{a\}\).
Page: 107
Chapter: 3
Section: 2.1
Variables: \(a\in \mathbb R\)
\(U(a)\) a neighborhood of \(A\)
Notation: \(\mathring U(a)\)
Index: 2
Kind: Def

Name: A (deleted) neighborhood of \(a\) in \(E\)
Definition: \(U(a)\cap E\) and \(\mathring U(a)\cap E\).
Page: 107
Chapter: 3
Section: 2.1
Variables: \(a\in \mathbb R\), \(U(a)\) a neighborhood of \(A\) and \(E\subseteq \mathbb R\)
Notation: \(U_E(a)\) and \(\mathring U_E(a)\)
Index: 2
Kind: Def

Smaller0: \(a\) is a limit point of \(E\)
Greater0: is not empty
Page: 111
Chapter: 3
Section: 2.2
Variables: \(a\in \mathbb R\),
\(U(a)\) a neighborhood of \(A\) and
\(E\subseteq\mathbb R\)
Index: B1)
Hide middle: yes
Prefix Greater: \(\mathring U_E(a)\)

Name: The limit of \(f\) as \(x\) tends to \(a\)
Definition: An element \(A\in Y\) such that, for all neighborhood \(V(A)\) of \(A\), there exists a deleted neighborhood \(\mathring U_ E(a)\) of \(a\) in \(E\) such that \(f(\mathring U_E(a))\subseteq V(A)\).
Page: 108
Chapter: 3
Section: 2.1
Variables: \(X,Y\) topological space
\(E\subseteq X\)
\(a\) a limit point of \(E\)
\(f:X\to Y\)
Notation: \[\lim_{\mathbb E\ni x\to a}f(x)\]
Kind: Def

Name: Signum of \(x\)
Definition: Sends negative to \(-1\), 0 to 0 and positive to \(1\).
Page: 108
Chapter: 3
Section: 2.1
Variables: \(x\) a real
Notation: \(\text{sgn }x\)
Index: 2
Kind: Ex

Question: \[\lim_{\mathbb R\setminus\{0\}\ni x\to 0}\text{sgn }x\]
Answer: does not exists.
Page: 108
Chapter: 3
Section: 2.1
Index: 2
Kind: Ex

Question: [$$]\lim_{\mathbb R\setminus\{0\}\ni x\to 0}\abs{\text{sgn }x}[/$$]
Answer: is 1.
Page: 109
Chapter: 3
Section: 2.1
Kind: Ex

Question: \[\lim_{\mathbb R_-\ni x\to 0}\text{sgn }x\]
Answer: is \(-1\).
Page: 109
Chapter: 3
Section: 2.1
Index: 4
Kind: Ex

Question: \[\lim_{\mathbb R\ni x\to 0}\sin \frac1x\]
Answer: does not exists.
Page: 109
Chapter: 3
Section: 2.1
Index: 5
Kind: Ex

Smaller0: \(\lim_{\mathbb E\ni x\to a}f(x)=A\)
Greater0: for each sequences \(\{x_n\}\) converging to \(a\), \(\lim_{n\to\infty}f(x_n)=A\)
Page: 110
Chapter: 3
Section: 2.1
Variables: \(E\subseteq \mathbb R\)
\(a\) a limit point of \(E\)
\(f:E\to\mathbb R\)
\(A\in\mathbb R\)
Index: 1
Kind: Prop
Smaller to greater: iff

Name: A real function ultimately constant as \(E\ni x\to a\)
Definition: \(f:E\to\mathbb R\) which is constant in some deleted neighborhood of \(a\).
Page: 110
Chapter: 3
Section: 2.2 a
Variables: \(E\subseteq \mathbb R\)
\(a\) a limit point of \(E\)
Index: 1
Kind: Prop

Name: A real function which is bounded (respectively above, below)
Definition: \(f:E\to\mathbb R\) such that there exists \(M\in\mathbb R\) such that [$]\abs{f(x)}<M[/$] (respectively, \(f(x)<M\), \(f(x)>M\).
Page: 110
Chapter: 3
Section: 2.2 a
Variables: \(E\subseteq \mathbb R\)
Index: 5
Kind: Def

Name: A real function which is bounded as \(E\ni x\to a\)
Definition: \(f:E\to\mathbb R\) such that there exists \(M\in\mathbb R\) such that [$]\abs{f(x)}<M[/$] on a deleted neighborhood of \(a\).
Page: 110
Chapter: 3
Section: 2.2 a
Variables: \(E\subseteq \mathbb R\),
\(a\) a limit point of \(E\)
Index: 5
Kind: Def

Name: A real function which is bounded below as \(E\ni x\to a\)
Definition: \(f:E\to\mathbb R\) such that there exists \(M\in\mathbb R\) such that \(f(x)>M\) on a deleted neighborhood of \(a\).
Page: 110
Chapter: 3
Section: 2.2 a
Variables: \(E\subseteq \mathbb R\),
\(a\) a limit point of \(E\)
Index: 5
Kind: Def

Name: A real function which is bounded above as \(E\ni x\to a\)
Definition: \(f:E\to\mathbb R\) such that there exists \(M\in\mathbb R\) such that \(f(x)<M\) on a deleted neighborhood of \(a\).
Page: 110
Chapter: 3
Section: 2.2 a
Variables: \(E\subseteq \mathbb R\),
\(a\) a limit point of \(E\)
Index: 5
Kind: Def

Smaller0: \(f\) is ultimately the constant \(A\) as \(E\ni x\to a\)
Greater0: \(\lim_{E\ni x\to a}f(x)=A\)
Page: 111
Chapter: 3
Section: 2.2 a
Variables: \(E\subseteq \mathbb R\)
\(a\) a limit point of \(E\)
\(f:E\to \mathbb R\)
\(A\in\mathbb R\)
Index: 1a)
Kind: Theo
Hide middle: yes

Smaller0: \(\lim_{E\ni x\to a}f(x)\) exists
Greater0: \(f\) is ultimately bounded as \(E\ni x\to a\)
Page: 111
Chapter: 3
Section: 2.2 a
Variables: \(E\subseteq \mathbb R\), \(a\) a limit point of \(E\) and \(f:E\to \mathbb R\) and \(A\in\mathbb R\)
Index: 1a)
Kind: Theo
Hide middle: yes

Question: \(\lim_{E\ni x\to a}f(x)=A_1\) and \(\lim_{E\ni x\to a}f(x)=A_2\) implies
Answer: \(A_1=A_2\).
Page: 111
Chapter: 3
Section: 2.2 a
Variables: \(E\subseteq \mathbb R\), \(a\) a limit point of \(E\) and \(f:E\to \mathbb R\) and \(A_1,A_2\in\mathbb R\)
Index: 1c)
Kind: Theo

Definition: \(x\mapsto f(x)+g(x)\)
Page: 112
Chapter: 3
Section: 2.2 b
Variables: \(E\subseteq\mathbb R\)
\(f,g:E\to\mathbb R\)
Notation: \(f+g\)

Definition: \(x\mapsto f(x) g(x)\)
Page: 112
Chapter: 3
Section: 2.2 b
Variables: \(E\subseteq\mathbb R\)
\(f,g:E\to\mathbb R\)
Notation: \(f\cdot g\)

Definition: \(x\mapsto f(x)/g(x)\)
Page: 112
Chapter: 3
Section: 2.2 b
Variables: \(E\subseteq\mathbb R\)
\(f,g:E\to\mathbb R\)
Notation: \(f/g\)
Assuming: \(g(x)\ne0\)

Definition: \(A+B\), \(AB\),\(A/B\) (assuming \(B\ne0\) and \(g(x)\ne0\))
Page: 112
Chapter: 3
Section: 2.2 b
Variables: \(E\subseteq\mathbb R\),
\(f,g:E\to\mathbb R\)
\(a\) a limit point of \(E\) such that \(\lim_{E\ni x\to a}f(x)=A\)
\(\lim_{E\ni x\to a}g(x)=B\).
Notation: \(\lim_{E\ni x\to a} f+g\), \(\lim_{E\ni x\to a} f\cdot g\), \(\lim_{E\ni x\to a}f/g\)

Name: A function infinitesimal at \(E\ni x\to a\)
Definition: A function \(f:E\to \mathbb R\) such that \(\lim_{E\ni x\to a}f(x)=0\).
Page: 112
Chapter: 3
Section: 2.2 b
Variables: \(E\subseteq\mathbb R\) and \(a\) a limit point of \(E\).

Smaller0: \(f,g\) are infinitesimal at \(E\ni x\to a\)
Greater0: \(f+g\) is infinitesimal at \(E\ni x\to a\)
Page: 112
Chapter: 3
Section: 2.2 b
Variables: \(E\subseteq\mathbb R\),
\(a\) a limit point of \(E\)
\(f,g:E\to\mathbb R\)
Index: 2a
Kind: Prop

Smaller0: \(f\) is infinitesimal and \(g\) bounded at \(E\ni x\to a\)
Greater0: infinitesimal at \(E\ni x\to a\)
Page: 112
Chapter: 3
Section: 2.2 b
Variables: \(E\subseteq\mathbb R\), \(a\) a limit point of \(E\), and \(f,g:E\to\mathbb R\)
Index: 2c
Kind: Prop
Prefix Greater: \(fg\) is

Question: \(\lim_{x\to 0}\frac{\sin x}{x}\)
Answer: is 1.
Page: 116
Chapter: 3
Section: 2.2 d
Index: 9
Kind: Ex

Name: \(a^x\)
Definition: if \(x=p/q\) then \(\sqrt[q]{a^p}\),
Page: 122
Chapter: 3
Section: 2.2 d
Variables: \(a>0\)
\(x\in\mathbb R\)
Index: 7
Kind: Def
Definition2: otherwise \(\lim_{\mathbb Q\ni y\to x}a^y\).
Conjdef: Either

Name: \(\log_ax\)
Definition: The inverse of \(x\mapsto a^x\).
Page: 122
Chapter: 3
Section: 2.2 d
Variables: \(a>0\) and \(x\in\mathbb R\)
Index: 7b
Kind: Def
Definition2: [$]\defFun{\mathbb R^{>0}}{\mathbb R}{x}{\frac{\log x}{\log a}}[/$]

Question: \(\log_a a\)
Answer: is 1
Page: 123
Chapter: 3
Section: 2.2 d
Variables: \(a>0\) and \(x\in\mathbb R\)
Index: 9 1')
Kind: Def

Question: \(\log_a (xy)\)
Answer: is \(\log_ax+\log_ay\)
Page: 123
Chapter: 3
Section: 2.2 d
Variables: \(a>0\) and \(x,y\in\mathbb R\)
Index: 9 1')
Kind: Def

Question: \(\lim_{\mathbb R_+\ni x\to y}\log_ax\)
Answer: is \(\log_a (y)\)
Page: 123
Chapter: 3
Section: 2.2 d
Variables: \(a>0\) and \(y\in\mathbb R\)
Index: 9 1')
Kind: Def

Smaller0: increasing
Greater0: \(a>1\)
Page: 123
Chapter: 3
Section: 2.2 d
Variables: \(a>0\)
\(a\ne 1\)
Index: 9 4')
Kind: Def
Smaller to greater: iff
Prefix Smaller: \(\log_a\) is

Question: The range of \(\log_{a}\), defined on \(\mathbb R^+\) is
Answer: \(\mathbb R\).
Page: 123
Chapter: 3
Section: 2.2 d
Variables: \(a>0\)
Index: 9 5
Kind: Def

Name: A (filter) base in \(X\)
Definition: such that for all \(B_1,B_2\) belonging to this set, there exists \(B\supseteq B_1\cap B_2\) belonging to this set
Page: 126, 128
Chapter: 3
Section: 2.3 a
Variables: \(X\) a set
Conjdef: A set of non-empty subsets of \(X\)
Denoted by: \(\mathcal B\)

Definition: \(\{x\in\mathbb R\mid x>a\}\), \(\{x\in\mathbb R\mid x<a\}\).
Page: 126
Chapter: 3
Section: 2.3
Variables: \(a\in\mathbb R\)
Notation: \(E_a^+\), \(E_a^-\)
Extra: This definition is not related to any particular set \(E\)

Name: \(x\) tends to \(a\) from the right
Page: 127
Chapter: 3
Section: 2.3
Variables: \(a\in\mathbb R\)
Notation: \(x\to a+0\)

Name: A limit of \(f\) over the base \(\mathcal B\)
Definition: A real \(A\) such that for all neighborhood \(V(A)\) of \(A\), there exists \(B\in \mathcal B\) such that \(F(B)\subseteq V(A)\).
Page: 127
Chapter: 3
Section: 2.3 b
Variables: \(X\) a set
\(\mathcal B\) a base in \(X\),
\(f:X\to\mathbb R\)
Notation: \(\lim_{\mathcal B} f(x)\)

Name: An ultimately constant function over the base \(\mathcal B\)
Definition: A function \(f:X\to\mathbb R\) such that there exists \(B\in\mathcal B\) such that \(F(B)\) is a singleton.
Page: 129
Chapter: 3
Section: 2.3 b
Variables: \(X\) a set and \(\mathcal B\) a base in \(X\),
Index: 13
Kind: Def

Name: An ultimately bounded function over the base \(\mathcal B\)
Definition: A function \(f:X\to\mathbb R\) such that there exists \(B\in\mathcal B\) such that \(F(B)\) is bounded.
Page: 129
Chapter: 3
Section: 2.3 b
Variables: \(X\) a set and \(\mathcal B\) a base in \(X\),

Name: An infinitesimal function over the base \(\mathcal B\)
Definition: A function \(f:X\to\mathbb R\) such that \(\lim_{\mathcal B}f=0\).
Page: 129
Chapter: 3
Section: 2.3 b
Variables: \(X\) a set
\(\mathcal B\) a base in \(X\)
Index: 15
Kind: Def

Name: The oscillation of \(f\) on \(E\)
Definition: The diameter of \(f(E)\).
Page: 130, 417
Chapter: 3
Section: 2.4 a
Variables: \(X\) a set,
\(f:X\to\mathbb R^n\) (in general: to a metric space),
\(E\subseteq X\)
Notation: \(\omega(f,E)\)
Index: 16, 5
Kind: Def, Def

Question: \(\omega(x^2,[-1,2])=\)
Answer: \(4\).
Page: 130
Chapter: 3
Section: 2.4 a
Index: 11
Kind: Ex

Question: \(\omega(x,[-1,2])=\)
Answer: \(3\).
Page: 130
Chapter: 3
Section: 2.4 a
Index: 11
Kind: Ex

Question: \(\omega(x,]-1,2[)=\)
Answer: \(3\).
Page: 130
Chapter: 3
Section: 2.4 a
Index: 11
Kind: Ex

Question: \(\omega(\text{sgn} x,[-1,2])=\)
Answer: \(2\).
Page: 130
Chapter: 3
Section: 2.4 a
Index: 11
Kind: Ex

Question: \(\omega(\text{sgn} x,[0,2])=\)
Answer: \(1\).
Page: 130
Chapter: 3
Section: 2.4 a
Index: 11
Kind: Ex

Question: \(\omega(\text{sgn} x,]0,2])=\)
Answer: \(0\).
Page: 130
Chapter: 3
Section: 2.4 a
Index: 16
Kind: Ex

Smaller0: \(f\) has a limit
Greater0: for each \(\epsilon>0\), there exists \(B\in\mathcal B\) such that \(\omega(f,B)<\epsilon\)
Page: 130, 418
Chapter: 3, 7
Section: 2.4 a, 2.1
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f:X\to\mathbb R^n\) (in general, to a complete space).
Index: 4, 1
Kind: Theo
Smaller to greater: iff
Name: The Cauchy criterion for the existence of a limit of a function

Name: The base \(x\to a\)
Definition: The set of deleted neighborhoods of \(a\).
Page: 17
Chapter: 3
Section: 2.3 a
Variables: \(a\in\mathbb R\)
Context: Analysis
Index: 1
Kind: Table

Definition: The set of deleted neighborhoods of \(a\) in \(E\).
Page: 17
Chapter: 3
Section: 2.3 a
Variables: \(E\) a set with limit point \(a\)
Notation: The base \(x\to a, x\in E\), or \(E\ni x\to a\)
Index: 1
Kind: Table
Notation2: \(x\to_{\in E}a\)

Smaller0: for every element \(B_Y\in\mathcal B_Y\) there exists \(B_X\in\mathcal B_X\) whose image \(f(B_X)\) is contained in \(B_Y\)
Greater0: \(\lim_{\mathcal B_X}(g\circ f)=\lim_{\mathcal B_Y}(g)\)
Page: 132, 417
Chapter: 3, 7
Section: 2.4 b, 2.1
Variables: \(X,Y\) sets with basis \(\mathcal B_X,\mathcal B_Y\).
\(f:X\to Y\),
\(g:Y\to \mathbb R^n\)
Index: 5, 2
Kind: Theo, Theo
Hide middle: yes
Assuming: Both limits are defined

Name: A monotonic function
Definition: A function which is either nonincreasing or nondecreasing.
Page: 135
Chapter: 3
Section: 2.4 c
Index: 17
Kind: Def

Name: An increasing/decreasing/nonincreasing/nondecreasing function
Definition: A function \(f:E\to\mathbb R\) with \(E\) ordered, such that \(x<y\) implies \(f(x)<, > ,\ge ,\le f(y)\).
Page: 135
Chapter: 3
Section: 2.4 c
Index: 17
Kind: Def

Name: A property of function which holds ultimately over \(\mathcal B\)
Definition: A property which holds over some \(B\in\mathcal B\).
Page: 137
Chapter: 3
Section: 2.4 d
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f:X\to\mathbb R\).
Index: 18
Kind: Def

Name: \(f\) is infinitesimal compared to \(g\) over the base \(\mathcal B\)
Definition: \(f=\alpha g\) with \(\alpha:X\to\mathbb R\) infinitesimal over \(\mathcal B\).
Page: 137
Chapter: 3
Section: 2.4 d
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f,g:X\to\mathbb R\).
Notation: \(f=_{\mathcal B}o(g)\)
Index: 19
Kind: Def

Name: An infinitesimal of higher order than \(g\) over \(\mathcal B\)
Definition: A function \(f=_{\mathcal B}o(g)\) with \(g\) infinitesimal over \(\mathcal B\).
Page: 137, 138
Chapter: 3
Section: 2.4 d
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\)
\(g:X\to\mathbb R\)
Index: 20, 26
Kind: Def, Ex
Example: \(x^{-2}\) as \(x\to+\infty\)
Assuming: \(g\) infinitesimal over \(\mathcal B\)

Name: An infinite function
Definition: A function that tends to infinity over \(\mathcal B\).
Page: 138
Chapter: 3
Section: 2.4 d
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\)
Index: 21
Kind: Def

Name: \(g\) is a higher order infinity than \(f\) over \(\mathcal B\)
Definition: \(f=_{\mathcal B}o(g)\).
Page: 138
Chapter: 3
Section: 2.4 d
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f,g:X\to\mathbb R\)
Index: 21
Kind: Def

Definition: \(f(x)=\beta(x)g(x)\) where \(\beta(x)\) is bounded over \(\mathcal B\).
Page: 139
Chapter: 3
Section: 2.4 d
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f,g:X\to\mathbb R\)
Notation: \(f=_{\mathcal B}O(g)\)
Index: 23
Kind: Def

Name: \(f\) and \(g\) are of the same order over \(\mathcal B\)
Definition: \(f=_{\mathcal B}O(g)\) and \(g=_{\mathcal B}O(f)\).
Page: 140
Chapter: 3
Section: 2 d
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f,g:X\to\mathbb R\)
Notation: \(f\asymp g\)
Index: 24
Kind: Def

Name: \(f\) and \(g\) are of the same order over \(\mathcal B\)
Definition: \(f=_{\mathcal B}O(g)\) and \(g=_{\mathcal B}O(f)\).
Page: 140
Chapter: 3
Section: 2 d
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f,g:X\to\mathbb R\)
Context: Analysis
Notation: \(f\asymp g\)
Index: 24
Kind: Def

Question: Is \((2+\sin x)x\) the same order as \(x\) for \(x\to\infty\).
Answer: Yes
Page: 140
Chapter: 3
Section: 2 d
Index: 34
Kind: Ex

Question: Is \((1+\sin x)x\) the same order as \(x\) for \(x\to\infty\).
Answer: No
Page: 140
Chapter: 3
Section: 2 d
Index: 34
Kind: Ex
Construction: \(1+\sin x\) is infinitely often 0

Name: \(f\) behaves asymptotically like \(g\) over \(\mathcal B\)
Definition: \(f=\gamma g\) with \(\lim_{\mathcal B}\gamma=1\)
Page: 140
Chapter: 3
Section: 2 d
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f,g:X\to\mathbb R\)
Notation: \(f\sim_{\mathcal B}g\)
Index: 25, 35, 38
Kind: Def, ex
Example: \(x^2\)
Name2: \(f\) is equivalent to \(g\) over \(\mathcal B\)
Example2: \(x\)
Typ: Equivalence relation
Applied to: \(x^2+x\sim_{x\to \infty}\)
Example4: \(\alpha x\)
Applied to3: \(\ln (1+x)\sim_{x\to 0}\)
Applied to2: \(\sin x\sim_{x\to 0}\)
Applied to4: \((1+x)^\alpha-1\sim_{x\to 0}\)

Q: \(e^x=_{x\to [...] }[...]+o([...])\).
A: 0; 1+x; x
Page: 141
Chapter: 3
Section: 2 d
Index: 39
Kind: Ex

Smaller0: \(\lim_{\mathcal B} fh\) or \(\lim_{\mathcal B} gh\) exists
Greater0: \(fh\sim_{\mathcal B}gh\)
Page: 142
Chapter: 3
Section: 2 d
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f,g,h:X\to\mathbb R\)
Kind: Prop
Smaller1: \(f\sim_{\mathcal B}g\)

Name: functions \(f,g:\mathbb R\to\mathbb R\) such that \(f\sim_{x\to\infty} g\) but \(\lim_{\to\infty}f-g\ne0\)
Page: 142
Chapter: 3
Section: 2 d
Index: 42
Kind: Ex
Example: \(\sqrt{x^2+x}\) and \(x\).

Question: \(o(f)+o(f)=\)
Answer: \(o(f)\).
Page: 142
Chapter: 3
Section: 2 d
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f:X\to\mathbb R\)
Index: 4a
Kind: Prop

Smaller0: \(g=o(f)\)
Greater0: \(g=O(f)\)
Page: 142
Chapter: 3
Section: 2 d
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f,g:X\to\mathbb R\)
Index: 4b
Kind: Prop

Question: \(o(f)+O(f)=\)
Answer: \(O(f)\).
Page: 142
Chapter: 3
Section: 2 d
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f:X\to\mathbb R\)
Index: 4c
Kind: Prop

Question: \(O(f)+O(f)=\)
Answer: \(O(f)\).
Page: 142
Chapter: 3
Section: 2 d
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f:X\to\mathbb R\)
Index: 4d
Kind: Prop

Question: \(\frac{o(f)}{g}=\)
Answer: \(o\left(\frac fg\right)\).
Page: 142
Chapter: 3
Section: 2 e
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f,g:X\to\mathbb R\) with \(g(x)\ne 0\) for all \(x\)
Index: 4d
Kind: Prop

Question: \(\frac{O(f)}{g}=\)
Answer: \(O\left(\frac fg\right)\).
Page: 142
Chapter: 3
Section: 2 e
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f,g:X\to\mathbb R\) with \(g(x)\ne 0\) for all \(x\)
Index: 4d
Kind: Prop

Name: A continuous function at \(a\)
Definition: A function defined on \(E\) such that for each neighborhood \(V(f(a))\) of \(f(a)\) there exists a neighborhood \(U_E(a)\) of \(a\) in \(E\) such that \(f(V(a))\subseteq U_E(a)\).
Page: 149, 421
Chapter: 4, 7
Section: 1.1, 2.2
Variables: \(E\subseteq\mathbb R^m\) a set and
\(a\in E\)
Index: 1, 6
Kind: Def
Definition2: The limit of \(f\) at \(a\) in \(E\) exists and equal \(f(a)\). The limit can be either on the metric space, or on the basis of deleted neighborhood.
Conjdef: A function \(f\) such that
Definition3: The limit of \(f\) on the set of neighborhoods of \(a\) exists.
Definition4: For each \(\epsilon>0\), there exists a neighborhood \(U_E(a)\) such that \(\omega(f,U_E(a))<\epsilon\).
Definition5: \(\omega(f,a)=0\)

Question: \(a\) is an isolated point implies
Answer: \(f\) is continuous at \(a\).
Page: 150
Chapter: 4
Section: 1.1
Variables: \(E\) a set,
\(a\in E\),
\(f:E\to \mathbb R\)
Index: 1
Kind: Def

Name: The oscillation of \(f\) at \(a\)
Definition: \(\lim_{\delta\to+0}\omega(f;U^\delta_E(a))\).
Page: 151, 422
Chapter: 4, 7
Section: 1.1, 2.2
Variables: \(E\subseteq\mathbb R^m\) a set,
\(a\in E\),
\(f:E\to\mathbb R^n\)
Notation: \(\omega(f;a)\)
Index: 2, 7
Kind: Def

Name: A function continuous on \(E\)
Definition: A function defined on \(E\) continuous on each point of \(E\).
Page: 152
Chapter: 4
Section: 1.1
Variables: \(E\) a set
Index: 3
Kind: Def

Name: The set of (real) function continuous on \(E\)
Page: 152
Chapter: 4
Section: 1.1
Variables: \(E\) a set
Notation: \(C(E)\)
Index: 3
Kind: Def
Notation2: \(C(E;\mathbb R)\)

Question: A constant function is
Answer: continuous.
Page: 152
Chapter: 4
Section: 1.1
Index: 1
Kind: Ex

Name: (A point of) discontinuity of \(f\)
Definition: An element \(a\in E\) such that \(f\) is not continuous at \(a\).
Page: 152
Chapter: 4
Section: 1.2
Variables: \(E\) a set
\(f:E\to\mathbb R\)
Kind: Def

Name: A discontinuity of first kind for \(f\)
Definition: A point \(a\in E\) such that \(\lim _{E\ni x\to a-0}f(x)\), \(\lim _{E\ni x\to a+0}f(x)\) exist, and one is not equal to \(f(a)\).
Page: 155
Chapter: 4
Section: 1.2
Variables: \(E\) a set,
\(f:E\to\mathbb R\)
Index: 6
Kind: Def

Name: A point of discontinuity of second kind for \(f\)
Definition: A point \(a\in E\) such that \(\lim _{E\ni x\to a-0}f(x)\) or \(\lim _{E\ni x\to a+0}f(x)\) does not exist.
Page: 156
Chapter: 4
Section: 1.2
Variables: \(E\) a set,
\(f:E\to\mathbb R\)
Index: 7
Kind: Def

Name: Dirichlet function
Definition: \(\mathbb R\to\{0,1\}\),
\(x\mapsto\left\{ \begin{array}{ll} 1,&\text{if }x\in \mathbb Q\\ 0,&\text{if }x\in\mathbb R\setminus\mathbb Q\end{array}\right.\)
Page: 156
Chapter: 4
Section: 1.2
Notation: \(\mathcal D\) function
Index: 11
Kind: Ex

Name: Riemann function
Definition: [$]\defFun{\mathbb R}{\mathbb R}{x}{\left\{\begin{array}{ll}\frac 1n,&\text{if }x=\frac{m}{n}\text{ is in lowest terms}\in \mathbb Q\\0,&\text{if }x\in\mathbb R\setminus\mathbb Q\end{array}\right.}[/$]
Page: 156
Chapter: 4
Section: 1.2
Notation: \(\mathcal R\) function
Index: 11
Kind: Ex

Question: The set of continuous point of the Riemann function is
Answer: \(\mathbb R\setminus\mathbb Q\).
Page: 156
https://en.wikipedia.org/wiki/Thomae's_function
Chapter: 4
Section: 1.2
Variables: [$]\defFun{\mathbb R}{\mathbb R}{x}{\left\{\begin{array}{ll}\frac 1n,&\text{if }x=\frac{m}{n}\text{ is in lowest terms}\in \mathbb Q\\0,&\text{if }x\in\mathbb R\setminus\mathbb Q\end{array}\right.}[/$] the Riemann function
Index: 12
Kind: Ex
Construction: The limit everywhere is 0. Indeed, for each \(N\in\mathbb N\) and \(\frac mn\in\mathbb N\) there exists a neighborhood of \(\frac mn\) where each rationals have denominator greater than \(N\)

Smaller0: \(f(a)\ne 0\)
Greater0: there is a neighborhood of \(a\) on which \(f\) have the sign of \(f\)
Page: 156
Chapter: 4
Section: 2.1
Variables: \(E\) a set,
\(a\in E\),
\(f:E\to\mathbb R\)
Index: 1 2
Kind: Theo
Hide middle: yes
Assuming: \(f:E\to\mathbb R\) continuous

Name: The set of function continuous at \(a\)
Page: 157
Section: 2.1
Chapter: 4
Index: 1 3
Variables: \(E\subseteq\mathbb R^n\) a set,
\(a\in E\)
Kind: Theo
Under: addition
Under2: multiplication
Under3: division,
Condition3: assuming the denominator is distinct from 0

Smaller0: \(f\) is continuous at \(a\) and \(g\) at \(f(a)\)
Greater0: \(g\circ f\) is continuous at \(a\)
Page: 157
Chapter: 4
Section: 2.1
Variables: \(E,Y\) a set,
\(f:E\to Y\), \(g:Y\to\mathbb R\),
\(a\in E\).
Index: 1 4
Kind: Theo
Hide middle: yes

Name: A local property of a function
Definition: A property of a function which is determined by the behavior of the function in an arbitrarily small neighborhood of the point in its domain of definition.
Page: 157
Chapter: 4
Section: 2.1
Index: 1 4
Kind: Theo

Name: A global property of a function
Definition: A property of a function which involves the entire domain of definition of the function.
Page: 157
Chapter: 4
Section: 2.1
Index: 1 4
Kind: Theo

Smaller0: \(E\) is a closed interval,
Greater0: \(f\) assumes the value 0
Page: 157
Chapter: 4
Section: 2.2
Variables: \(E\subseteq\mathbb R\),
\(f:E\to\mathbb R\)
Index: 2
Kind: Theo
Hide middle: yes
Smaller1: \(f\) continuous which assumes values with different signs at the endpoints of the interval
Name: The Bolzano-Cauchy intermediate-value theorem

Smaller0: \(E\) is an open interval
Greater0: \(f\) assumes all values of \([A,B]\)
Page: 159
Chapter: 4
Section: 2.2
Variables: \(E\subseteq\mathbb R\),
\(f:E\to\mathbb R\),
\(A<B\)
Index: 2
Kind: Cor
Hide middle: yes
Smaller1: \(f\) continuous which assumes values \(A\) and \(B\)

Smaller0: \(E\) is a closed interval
Greater0: \(f\) is bounded
Page: 159
Chapter: 4
Section: 2.2
Variables: \(E\subseteq\mathbb R\),
\(f:E\to\mathbb R\)
Index: 3
Kind: Theo
Hide middle: yes
Smaller1: \(f\) continuous
Name: Weierstrass maximum-value theorem

Name: An uniformly continuous function
Definition: A function \(f:E\to\mathbb R^n\) such that, for every \(\epsilon>0\), there exists \(\delta>0\) such that \(d(f(x),f(y))<\epsilon\) for all \(x,y\) such that \(d(x,y)<\delta\).
Page: 159, 423, 174
Chapter: 4, 7, 3 munkres
Section: 2.2, 22.2, 27,
Variables: \(E\subseteq\mathbb R^m\)
Index: 3, 8
Kind: Def, ,def
Example: Continuous functions on compacts
Example2: Affine functions
Counterexample: \(x^2\)
Counterexample2: \(\sin x^2\)

Smaller0: \(f\) is unbounded in every neighborhood of a fixed point \(x_0\in E\)
Greater0: \(f\) is not uniformly continuous
Page: 161
Chapter: 4
Section: 2.2
Variables: \(E\subseteq\mathbb R\),
\(f:E\to\mathbb R\)
Index: 5
Kind: Ex
Construction: take \(\delta>0\), there are points \(x_1,x_2\) in the \(\delta/2\)-neighborhood of \(x_0\) such that [$]\abs{f(x_1)-f(x_2)}>1[/$] although [$]\abs{x_1-x_2}<\delta[/$].
Counterexample: Exponential

Smaller0: \(E\) is a closed interval
Greater0: \(f\) is uniformly continuous
Page: 161
Chapter: 4
Section: 2.2
Variables: \(E\subseteq\mathbb R\)
\(f:E\to\mathbb R\)
Kind: Theo
Hide middle: yes
Smaller1: \(f\) continuous
Name: The Cantor-Heine theorem on uniform continuity
Extra: counter example ?
Connect Smaller0: and

Smaller0: \(f\) is strictly monotonic
Greater0: \(f\) has an inverse on \(f(X)\)
Page: 166
Chapter: 4
Section: 2.2
Variables: \(X\subseteq\mathbb R\)
\(f:X\to\mathbb R\)
Index: 5
Kind: Theo
Name: Inverse function theorem

Name: \(f^{-1}\)
Page: 163
Chapter: 4
Section: 2.2
Variables: \(X\subseteq\mathbb R\)
\(f:X\to\mathbb R\) strictly monotonic
Kind: Prop
Typ: Monotonic of the same kind

Smaller0: \(f\) is strictly monotonic and continuous
Greater0: equals \([f(a),f(b)]\)
Page: 166
Chapter: 4
Section: 2.2
Variables: \(a<b\in\mathbb R\)
\(f:[a,b]\to\mathbb R\) monotone
Index: 5
Kind: Theo
Prefix Greater: \(f([a,b])\)
Greater1: continuous
Connect Greater0: and \(f^{-1}\) is

Name: A divergent series
Definition: A series \(\sum a_i\) such that \(s_i\) diverges.
Page: 95
Chapter: 3
Section: 1.4 a
Index: 19
Kind: Def

Name: \(\{x_n\}\) tends to positive infinity
Definition: For each \(c\in\mathbb R\), \(x_i>c\) for \(i\) great enough.
Page: 91
Chapter: 3
Section: 1.3.d
Variables: \(\{x_N\}\in\mathbb R\) a sequence
Notation: \(x_n\to+\infty\)
Index: 12
Kind: Def

Name: \(\{x_n\}\) tends to negative infinity
Definition: For each \(c\in\mathbb R\), \(x_i<c\) for \(i\) great enough.
Page: 91
Chapter: 3
Section: 1.3.d
Variables: \(\{x_N\}\in\mathbb R\) a sequence
Notation: \(x_n\to-\infty\)
Index: 12
Kind: Def

Definition: The set of neighborhoods of infinity in \(E\).
Page: 127
Chapter: 3
Section: 2.3 a
Variables: \(E\) an unbounded set
Notation: The base \(x\to \infty, x\in E\)
Index: 1
Kind: Table
Notation2: \(x\to_{\in E}\infty\)
Notation3: \(E\ni x\to\infty\)

Definition: \(\mathbb R^m\setminus B(a;r)\) for each \(a\in\mathbb R^m\), \(r>0\).
Page: 17, 418
Chapter: 3, 7
Section: 2.3 a, 2.1
Variables: \(m\ge0\)
Notation: The base \(x\to\infty\)
Index: 1
Kind: Table
Extra: "The norm function" was in Example. Try to understand why

Name: A removable point of discontinuity
Definition: A point of discontinuity \(\xi\in\Omega\) such that there exists \(g:E\to\mathbb R\) continuous, such that \(f=g\) on \(\Omega\subseteq\{\xi\}\)
Page: 155,
Chapter: 4, 4
Section: 1.2, 3.1
Variables: \(\omega\subseteq\mathbb R^n\) a region
\(f:\omega\setminus\{\xi\}\to\mathbb C\)
Index: 4
Kind: Def
Example: 0
Example2: any isolated point of \(X\)
Extra: There are equivalent definition for complex holomorphic function
Applied to: the dirac function
Applied to2: the characteristic function of \(X\)

Smaller0: \(\{x_i\}\) converges to \(A\)
Greater0: \(V\) contains all but finitely many elements of the sequence
Page: 82
Chapter: 3
Section: 1.2.a
Variables: \(\{x_i\}\) a sequence,
\(A\in\mathbb R\) and
\(V\) a neighborhood of \(A\)
Index: 1 b)
Kind: Theo

Smaller0: \(\{x_n\}\) is a Cauchy sequence
Greater0: \(\{x_n\}\) converges
Page: 85
Chapter: 3
Section: 1.3.a
Variables: \(\{x_n\}\in\mathbb R\)
Index: 4
Kind: Theo
Smaller to greater: iff
Context: Real analysis

Smaller0: [$]\overline\lim_{n\to \infty}\abs{\frac{a_{n+1} }{a_n}}<1[/$]
Greater0: \(\sum a_i\) converges absolutely
Page: 100
Chapter: 3
Section: 1.4 b
Variables: \(\sum a_i\) series
Index: 10
Kind: Cor
Name: D'Alembert's test
Construction: take [$]q\in(\overline\lim_{n\to \infty}\abs{\frac{a_{n+1} }{a_n}},1)[/$]. Ignore the first terms, to assume that [$]\abs{\frac{a_{n+1}}{a_n}}<1[/$] for all \(n\). Then [$]\abs{a_{n+1}}\le\abs{a_1}q^n[/$], thus [$]\sum\abs{a_i}\le\sum_{n=1}^\infty\abs{a_1}q^n=\frac{\abs{a_1}q}{1-q}[/$], it converges absolutely

Smaller0: [$]\underline\lim_{n\to \infty}\abs{\frac{a_{n+1} }{a_n}}>1[/$]
Greater0: \(\sum a_i\) diverges
Page: 100
Chapter: 3
Section: 1.4 b
Variables: \(\sum a_i\) series
Index: 10
Kind: Cor
Name: D'Alembert's test
Construction: take [$]q\in(1,\overline\lim_{n\to \infty}\abs{\frac{a_{n+1} }{a_n}})[/$]. Ignore the first terms, to assume that [$]\abs{\frac{a_{n+1}}{a_n}}>1[/$] for all \(n\). Then [$]\abs{a_{n+1}}\ge\abs{a_1}q^n[/$] does not tend toward 0

Smaller0: \(\lim_{E\ni x\to a}f(x)<\lim_{E\ni x\to a}g(x)\)
Greater0: there exists a deleted neighborhood \(\mathring U(a)\) of \(a\) on which \(f<g\)
Page: 112
Chapter: 3
Section: 2.2 c
Variables: \(E\subseteq\mathbb R\),
\(f,g:E\to\mathbb R\) and
\(a\) a limit point of \(E\)
Index: 3a
Kind: Theo

Smaller0: \(\{x_n\}\) has a converging subsequence
Greater0: \(\{x_n\}\) has a subsequence that tends to infinity
Page: 91
Chapter: 3
Section: 1.3.d
Variables: \(\{x_n\}\in\mathbb R\) a sequence
Index: 2
Kind: Lem
Smaller to greater: or

Smaller0: \(f=_{\mathcal B}O(1)\)
Greater0: \(f\) is ultimately bounded over \(\mathcal B\)
Page: 140
Chapter: 3
Section: 2.4 d
Variables: \(X\) a set,
\(\mathcal B\) a base in \(X\),
\(f:X\to\mathbb R\)
Index: 23
Kind: Def
Smaller to greater: iff

Smaller0: \(A=C\)
Greater0: \(C=B\)
Page: 112
Chapter: 3
Section: 2.2 c
Variables: \(E\subseteq\mathbb R\),
\(f,g,h:E\to\mathbb R\)
\(a\) a limit point of \(E\) such that \(\lim_{E\ni x\to a}f(x)=A\), \(\lim_{E\ni x\to a}g(x)=B\) and \(\lim_{E\ni x\to a}h(x)=C\).
Index: 3b
Kind: Theo
Hide middle: yes
Smaller1: \(f\le g\le h\)

Smaller0: \(f\ge g\)
Greater0: \(A\ge B\)
Page: 112
Chapter: 3
Section: 2.2 c
Variables: \(E\subseteq\mathbb R\),
\(f,g,h:E\to\mathbb R\)
\(a\) a limit point of \(E\) such that \(\lim_{E\ni x\to a}f(x)=A\) and \(\lim_{E\ni x\to a}g(x)=B\)
Kind: Cor
Hide middle: yes
Smaller1: \(f\ge B\)
Connect Smaller0: or
Suffix Smaller: on a deleted neighborhood of \(a\)

Smaller0: \(\{x_n\}\) is bounded
Greater0: \(\{x_n\}\) contains a converging subsequence
Page: 90
Chapter: 3
Section: 1.3.d
Variables: \(\{x_N\}\in\mathbb R\) a sequence
Index: 1
Kind: Lem
Name: Bolzano Weierstrass

Question: \(\{x_n\}+\{y_n\}\) converges to
Answer: The limit of \(\{x_n+y_n\}\) if it exists
Page: 82
Chapter: 3
Section: 1.2.b
Variables: \(\{x_n\}\), \(\{y_n\}\) sequences with limit \(A\) and \(B\)
Index: 2
Kind: Theo

Question: \(\{x_n\}/\{y_n\}\) converges to
Answer: The limit of \(\{x_n/y_n\}\) if it exists
Page: 82
Chapter: 3
Section: 1.2.b
Variables: \(\{x_n\}\), \(\{y_n\}\) sequences with limit \(A\) and \(B\)
Index: 2
Kind: Theo

Question: \(\{x_n\}\times \{y_n\}\) converges to
Answer: The limit of \(\{x_n\times y_n\}\) if it exists
Page: 82
Chapter: 3
Section: 1.2.b
Variables: \(\{x_n\}\), \(\{y_n\}\) sequences with limit \(A\) and \(B\)
Index: 2
Kind: Theo

Smaller0: \(f\) have a limit at \(\text{sup} E\)
Greater0: \(f\) is bounded above
Page: 136
Chapter: 3
Section: 2.4 c
Variables: \(E\) an ordered set,
\(f:E\to\mathbb R\)
Index: 6
Kind: Theo
Smaller to greater: iff
Smaller1: \(f\) non decreasing

Smaller0: \(f\) is injective
Greater0: \(f\) is monotonic
Page: 163
Chapter: 4
Section: 2.2
Index: 1
Kind: Prop
Smaller to greater: iff
Assuming: \(f\) is continuous
Context: real analysis

Question: \(\lim_{x\to+0}x^\alpha\log_{a}(x)\)
Answer: \(o(1)\)
Page: 139
Chapter: 3
Section: 2.4 d
Variables: \(\alpha>0\), \(a>0\)
Context: \(o()\)
Index: 32
Kind: Ex