Title: Theory / Unimportant / Math / Analysis / Real Pugh/{ / 3 - Function of a real variable
Tags: fromMath mathToDef complex automaticImplies riemanIntegralOfF closed mathToSee récupéré checkDeck examplesSwitchExample1 hasSeenConjDef integralprimitive multipartite onMultipleBook

Name: A function \(\mathbb R\to\mathbb R\) differentiable at \(x\in (a,b)\)
Definition: \(f:(a,b)\to \mathbb R\) such that \(\lim_{t\to x}\frac{f(t)-f(x)}{t-x}\) exists.
Page: 149
Chapter: 3
Section: 1
Variables: \(a<b\) three reals.

Name: \(\Delta f(x)\)
Definition: \(f(t)-f(x)\)
Page: 149
Chapter: 3
Section: 1
Variables: \(a{\lt}x{\lt}b\) three reals
\(f:(a,b)\to \mathbb R\).
Context: Analysis

Definition: \(\lim_{\Delta x\to 0}\frac{\Delta f}{\Delta x}\).
Page: 149
Chapter: 3
Section: 1
Variables: \(a{\lt}x{\lt}b\) three reals
\(f:(a,b)\to \mathbb R\).
Notation: \(f'(x)\)

Name: A function which satisfies the Global Lipschitz condition
Definition: [$]\abs{f(t)-f(x)}\le M\abs{t-x}[/$] for some \(M>0\).
Page: 152
Chapter: 3
Section: 1
Variables: \(f:[a,b]\to \mathbb R\).
Index: 5
Kind: Cor

Name: A smooth function
Definition: A function which have \(r\)-th derivative for all \(r\in\mathbb N\).
Page: 157
Chapter: 3
Section: 1

Definition: \(\bigcap_{i\in\mathbb N}C^i\).
Page: 158
Chapter: 3
Section: 1
Context: Analysis
Notation: Class \(C^\infty\)

Name: An analytic function on \((a,b)\)
Definition: A function \(f:(a,b)\to \mathbb R\) such that for each \(x\in(a,b)\) there exists \(\delta>0\) and a power series \(\sum a_rh^r\) such that if \(|h|<\delta\) then the series converges and \(f(x+h)=\sum_{r=0}^{\infty}a_rh^r\).
Page: 158
Chapter: 3
Section: 1
Variables: \(a<b\) two reals.

Definition: The class of analytic functions
Page: 158
Chapter: 3
Section: 1
Notation: \(C^\omega\)

Name: Smooth non-analytic function
Page: 159
Chapter: 3
Section: 1
Example: \(e(x)=\left\{\begin{array}{ll}\exp^{-1/x}&\text{if }x>0\\0&\text{if }x\le 0\end{array}\right.\).

Name: The \(r\)th Taylor polynomial of \(f\) at \(x\)
Definition: \(h\mapsto\sum_{i=0}^r \frac{f^{(i)}(x)}{i!}h^i\).
Page: 159
Chapter: 3
Section: 1
Variables: \(a<b\) reals and
\(f:(a,b)\to \mathbb R\).

Name: A function which is \(r\)th order flat at \(0\).
Definition: \(f:(a,b)\to \mathbb R\) such that \(\frac{f(x)}{x^r}\to 0\) as \(x\to 0\).
Page: 160
Chapter: 3
Section: 1
Variables: \(a<0<b\) two reals.
Index: 12
Kind: Theo

Name: A \(C^r\) diffeomorphism
Definition: An homeomorphism \(f\) such that \(f\) and \(f^{-1}\) are \(C^r\).
Page: 163
Chapter: 3
Section: 1

Name: The mesh of \(P\)
Definition: The length of the largest subinterval \([x_{i-1},x_i]\).
Page: 164
Chapter: 3
Section: 2
Variables: \(a<b\) two reals,
\(P\) a partition of \([a,b]\).
Context: Partition
Name2: The norm of \(P\)
French: Le pas de \(P\)

Name: Riemann integral of \(f\) over \([a,b]\)
Definition: a real \(I\) such that, for all \(\epsilon>0\), there exists \(\delta>0\) such that for all \(P,T\) partition pair of mesh less that \(\delta\), \(|R(f,P,T)-I|<\epsilon\).
Page: 164
Chapter: 3
Section: 2
Variables: \(a<b\) two reals
\(f:[a,b]\to\mathbb R\)
Notation: [$]\int_a^bf(x)\dd x[/$]
Definition2: \(\overline {\mathcal J}\) and \(\underline {\mathcal J}\) if they are equal
Extra: Real Pugh, Ch 3 S2 p164
Visual Complex  Ch 8II, S 1, P 378
Without the "partition pair", it can be stated as The limit of \(\sum_{i=1}^nf(x_i)\Delta_i\), when \(a=p_0<x_1<p_1<\dots<p_{n-1}<x_n<p_n=b\), \(\Delta_i=p_i-p_{i-1}\), and \(\max\Delta_i\) tends to 0

Name: The lower sum of \(f\) with respect to a partition \(P\) of \([a,b]\)
Definition: \[\sum_{i=1}^n\text{inf}\{f(t):x_{i-1}\le t\le x_i\}\Delta x_i\].
Page: 166
Chapter: 3
Section: 2
Variables: \(a<b\) two reals,
\(f:[a,b]\to\mathbb R\)
\(P\) a partition of \([a,b]\).
Notation: \(L(f,P)\)

Name: Darboux integral of \(f\) on \([a,b]\)
Definition: The lower and upper integral of \(f\) on \([a,b]\) if they are equal.
Page: 167
Chapter: 3
Section: 2
Variables: \(a<b\) two reals
\(f:[a,b]\to\mathbb R\)

Name: The characteristic function of \(E\)
Definition: \(\mathcal X_E(x)\) is \(1\) if \(x\in E\) and 0 otherwise.
Page: 171
Chapter: 3
Section: 2
Variables: \(E\subseteq S\) sets.
Notation: \(\mathcal X_E:S\to \{0,1\}\)

Name: Step functions
Definition: Finite sum of constants times characteristic functions of intervals.
Page: 172
Chapter: 3
Section: 2
Kind: Chapitre
Property: Riemann integrable

Name: Piecewise continuous function
Definition: A function which is continuous, except at finitely many points.
Page: 172
Chapter: 3
Section: 2

Name: The Rational ruler function
Definition: A function \(f:\mathbb R\to\mathbb R\) such that \(f(p/q)=1/q\) and \(f(x)=0\) for \(x\not\in\mathbb Q\).
Page: 174
Chapter: 3
Section: 2

Name: The class of zero set
Page: 176
Section: 2
Chapter: 3
Extra: of r? product?
Under: union
Condition: countable
Under2: taking subset.

Smaller0: \(f\) is bounded piecewise continuous
Greater0: \(f\) is a Riemann integrable function
Page: 179
Chapter: 3
Section: 2
Variables: \(a<b\) two reals
\(f:[a,b]\to\mathbb R\).
Index: 24, 25, 26
Kind: Cor
Smaller1: \(f\) is a characteristic function of \(S\subseteq [a,b]\) whose boundary is a zero set
Connect Smaller0: or
Smaller2: \(f\) is a monotone functions

Name: An antiderivative of \(f\)
Definition: A function \(g\) such that \(g'=f\).
Page: 184
Chapter: 3
Section: 2
Variables: \(a<b\) two reals,
\(f:[a,b]\to\mathbb R\).
Extra: Real analysis, Pug, Ch3.2, P 184

Name: The Devil's staircase function
Definition: to \(x\) if no such \(w_i\) is 1
Page: 186
Chapter: 3
Section: 2
Index: 37
Kind: Theorem
Definition2: \(w_1\dots w_n\) if \(n+1\) is the least integer such that \(w_{n+1}=2\)
Conjdef: The function \([0,1]\to[0,1]\) sending \(x=.w_1w_2\dots\) in base 3
Extra: Example of: Continuous non-constant function whose derivative is 0 almosts everywhere

Name: A series
Definition: A formal sum \(\sum a_k\).
Page: 191
Chapter: 3

Name: The \(n\)-th partial sum of \(\sum a_i\)
Definition: \(\sum_{i=0}^na_i\).
Page: 191
Chapter: 3
Variables: \(\sum a_i\) a series.

Name: Divergent series
Definition: A series \(\sum_{i=0}^n a_i\) such that \(s_n\) does not converges.
Page: 191
Chapter: 3

Smaller0: \(|a_k|\)
Greater0: \(\sum b_k\) converges implies \(\sum a_k\) converges
Page: 192
Chapter: 3
Variables: \(\sum a_i,\sum b_i\) series.
Index: 40
Name: Comparison test

Smaller0: convergence of the improper integrals
Greater0: convergence of the series
Page: 193
Chapter: 3
Variables: \(\sum a_i\) a serie and
\(f:\mathbb R^{\ge0}\to\mathbb R\).
Suppose that \(\sum_0^\infty f(x)dx\) is a given improper integral and \(\sum a_k\) is a given series.
Index: 41
Assuming: \(|a_k|\le f(x)\) for all sufficiently large \(k\) and all \(x\in(k-1,k]\)

Smaller0: divergence of the improper integrals
Greater0: divergence of the series
Page: 193
Chapter: 3
Variables: \(\sum a_i\) a serie and
\(f:\mathbb R^{\ge0}\to\mathbb R\).
Index: 41
Assuming: \(|f(x)|\le a_k\) for all sufficiently large \(k\) and all \(x\in[k,k+1)\)

Name: The exponential growth rate of a serie \(\sum a_k\)
Definition: \(\lim\sup_{k\to \infty}\sqrt[k]{|a_k|}\).
Page: 194
Chapter: 3
Variables: \(\sum a_i\) a series.
Denoted by: \(\alpha\)

Name: The lower integral of \(f\) on \([a,b]\)
Definition: \(\text{sup}_PL(f,P)\).
Page: 166
Chapter: 3
Section: 2
Variables: \(a<b\) two reals
\(f:[a,b]\to\mathbb R\).

Name: The upper integral of \(f\) on \([a,b]\)
Definition: \(\text{inf}_PU(f,P)\).
Page: 166
Chapter: 3
Section: 2
Variables: \(a<b\) two reals
\(f:[a,b]\to\mathbb R\).

Name: Convergent series
Definition: A series \(\sum_{i=0}^n a_i\) such that \(s_n\) converges.
Page: 191
Chapter: 3

Name: The upper sum of \(f\) with respect to a partition \(P\) of \([a,b]\)
Definition: \[\sum_{i=1}^n\text{sup}\{f(t):x_{i-1}\le t\le x_i\}\Delta x_i\].
Page: 166
Chapter: 3
Section: 2
Variables: \(a<b\) two reals,
\(f:[a,b]\to\mathbb R\)
\(P\) a partition of \([a,b]\).
Notation: \(U(f,P)\)

Name: A partition pair of the segment \([a,b]\)
Definition: two finite sets of points \(P,T\subseteq[a,b]\) where \(P=\{x_0,\dots,x_n\}\) and \(T=\{t_1,\dots,t_n\}\) are interlaced as \(a=x_0\le t_1\le x_1\le\dots \le t_n\le x_n=b\). and the \(x_i\) distincts.
Page: 164
Chapter: 3
Section: 2
Variables: \(a<b\) two reals.

Name: A zero set
Definition: A set \(M\subseteq \mathbb R\) such that, for each \(\epsilon>0\), there is a countable covering of \(M\) by open intervals \((a_i,b_i)\) such that \(\sum b_i-a_i\le\epsilon\).
Page: 175, 176
Chapter: 3
Section: 2, 2
Example: At most countable set
Example2: The middle-thirds Cantor set.

Name: Riemann's Integrability Criterion
Definition: \(f\) is Riemann integrable iff \(\forall\epsilon>0\exists P\) such that \(U(f,P)-L(f,P)<\epsilon\)
Page: 171
Chapter: 3
Section: 2
Variables: \(a<b\) two reals,
\(f:[a,b]\to\mathbb R\) bounded.
Index: 21

Name: A Riemann integrable function
Definition: \(\forall\epsilon>0\exists P\) such that \(U(f,P)-L(f,P)<\epsilon\)
Page: 171
Chapter: 3
Section: 2
Variables: \(a<b\) two reals,
Index: 21
Conjdef: A function \(f:[a,b]\to\mathbb R\) such that

Question: \((f+g)'=\)
Answer: \(f'+g'\)
Page: 149
Chapter: 3
Section: 1
Variables: \(a<x<b\) three reals and \(f,g:(a,b)\to \mathbb R\).
Index: 1b

Question: \((f\times g)'(x)=\)
Answer: \(f(x)\times g'(x)+f'(x)\times g(x)\).
Page: 149
Chapter: 3
Section: 1
Variables: \(a<x<b\) three reals
\(f,g:(a,b)\to \mathbb R\)
Index: 1c

Smaller0: \(f\) is constant
Greater0: \(0\)
Page: 150
Chapter: 3
Section: 1
Variables: \(f:(a,b)\to\mathbb R\)
Index: 1d
Hide middle: yes
Prefix Greater: \(f'=\)

Question: \(\left(\frac fg\right)'(x)=\)
Answer: \(\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2} \)
Page: 150
Chapter: 3
Section: 1
Variables: \(a<x<b\) three reals and \(f:(a,b)\to \mathbb R\).
Index: 1e

Smaller0: \(\left(g\circ f\right)'(x)\)
Greater0: \(g'(f(x))f'(x)\)
Page: 150
Chapter: 3
Section: 1
Variables: \(f,g:(a,b)\to\mathbb R\)
\(x\in (a,b)\).
Index: 1f
Smaller to greater: equals
Hide middle: yes
Assuming: \(f\) is differentiable at \(x\) and \(g\) at \(f(x)\)

Question: \(\left(\sum_{i=0}^{n}a_ix^i\right)'=\)
Answer: \(\sum_{i=0}^{n-1}(i+1)a_{i+1}x^i\)
Page: 151
Chapter: 3
Section: 1
Variables: \(a_0,\dots,a_n\in\mathbb R\).
Index: 2
Kind: Cor

Question: \(f:(a,b)\to\mathbb R\) is differentiable and achieves a min or max at some \(\theta\in(a,b)\) implies \(f'(\theta)=\)
Answer: \(0\)
Page: 151
Chapter: 3
Section: 1
Variables: \(a<x<b\) three reals and \(f:(a,b)\to \mathbb R\).
Index: 4
Kind: Lem

Smaller0: \(f\) is differentiable on \((a,b)\)
Greater0: its derivative function \(f'(x)\)
Page: 154
Chapter: 3
Section: 1
Variables: \(f:[a,b]\to \mathbb R\).
Index: 8
Kind: Theo
Hide middle: yes
Greater1: the intermediate value property
Connect Greater0: has

Question: \(f'\)
Answer: do not have any jump discontinuity.
Page: 155
Chapter: 3
Section: 1
Variables: \(a<x<b\) three reals
\(f:(a,b)\to \mathbb R\) differentiable.
Index: 9
Kind: Cor

Question: \(f^{(r)}(x)=\)
Answer: \(\left(f^{(r-1)}\right)'\).
Page: 157
Chapter: 3
Section: 1
Variables: \(a<x<b\) three reals
\(r\in\mathbb N^{>0}\)
\(f:(a,b)\to \mathbb R\)

Question: If \(f\) is \(r^{\text{th}}\)-order differentiable and \(r\ge 1\) then \(f^{(r-1)}(x)\) is
Answer: continuous.
Page: 157
Chapter: 3
Section: 1
Variables: \(a<x<b\) three reals and \(f:(a,b)\to \mathbb R\).
Index: 10
Kind: Theo

Name: the \(r\)th Taylor approximation of \(f\)
Definition: the only polynomial \(P\) of degree at most \(r\) such that \(f-P\) is \(r\)-th order flat.
Page: 160
Chapter: 3
Section: 1
Variables: \(f:(a,b)\to\mathbb R\).
Notation: \(P\)
Index: 12
Kind: Theo
Assuming: \(f\) is \(r\)-th order differentiable

Definition: \(h\mapsto f(x+h)-P(h)\)
Page: 160
Chapter: 3
Section: 1
Variables: \(f:(a,b)\to\mathbb R\).
Notation: \(R\)
Index: 12 (a), (c)
Kind: Theo
Typ: \(r\)th order flat at \(h=0\)
Assuming: \(f\) is \(r\)-th order differentiable

Question: \(\left(f^{-1}\right)'(x)=\)
Answer: \(\frac1{f'\circ f^{-1}(y)}\).
Page: 162
Chapter: 3
Section: 1
Variables: \(a{\lt}b\) reals
\(f:(a,b)\to \mathbb R\).
Index: 14
Kind: Theo
Assuming: \(f\) is a differentiable surjection, and \(f'(x)\ne 0\) for all \(x\in(a,b)\)

Definition: The Riemann sum corresponding to \(f\), \(P\) and \(T\).
Page: 164
Chapter: 3
Section: 2
Variables: \(a<b\) two reals,
\(f:[a,b]\to\mathbb R\)
\(P,T\) a partition pair of \([a,b]\).
Notation: \(R(f,P,T)\)

Name: The Riemann sum corresponding to \(f,P,T\)
Definition: \(\sum_{i=1}^nf(t_i)\Delta x_i\) with \(\Delta x_i=x_i-x_{i-1}\).
Page: 164
Chapter: 3
Section: 2
Variables: \(a<b\) two reals, \(f:[a,b]\to\mathbb R\) and \(P,T\) a partition pair of \([a,b]\).

Smaller0: \(f\) is Riemann integrable
Greater0: \(f\) is bounded
Page: 165
Chapter: 3
Section: 2
Variables: \(a<b\) two reals, \(f:[a,b]\to\mathbb R\).
Index: 16
Kind: Theo

Question: Algebraically, the integral function is
Answer: linear.
Page: 165
Chapter: 3
Section: 2
Index: 17
Kind: Theo

Smaller0: \(f\le g\)
Greater0: [$]\int_a^bf(x)\dd x[/$]
Page: 166
Chapter: 3
Section: 2
Variables: \(a<b\) two reals
\(f,g:[a,b]\to\mathbb R\).
Index: 18
Kind: Theo
Smaller1: \(f,g\in\mathcal R\)
Greater1: [$]\int_a^bg(x)\dd x[/$]
Connect Greater0: \(\le\)

Smaller0: \(|f|\le M\)
Greater0: [$]|\int_a^b f(x)\dd x|[/$]
Page: 166
Chapter: 3
Section: 2
Variables: \(a<b\) two reals
\(f:[a,b]\to\mathbb R\).
Index: 19
Kind: Cor
Smaller1: \(f\in\mathcal R\)
Greater1: \(M(b-a)\)
Connect Greater0: \(\le\)

Smaller0: \(f\)'s Darboux integral equal \(r\)
Greater0: \(f\)'s Riemann integral equal \(r\)
Page: 167
Chapter: 3
Section: 2
Variables: \(a<b\) two reals
\(r\in\mathbb R\)
\(f:[a,b]\to\mathbb R\) Darboux and Riemann's integral
Index: 20
Kind: Theo
Smaller to greater: iff
Hide middle: yes

Smaller0: \((P,T)\) refines \((P',T')\)
Greater0: greater
Page: 168
Chapter: 3
Section: 2
Variables: \(a<b\) two reals,
\(f:[a,b]\to\mathbb R\).
\((P,T), (P',T')\) two partitions of \(a,b\).
Hide middle: yes
Prefix Greater: the lower sum according to \((P,T)\) is
Suffix Greater: than the one according to \((P',T')\)

Smaller0: \((P,T)\) refines \((P',T')\)
Greater0: smaller
Page: 168
Chapter: 3
Section: 2
Variables: \(a<b\) reals,
\(f:[a,b]\to\mathbb R\)
\((P,T), (P',T')\) two partitions of \(a,b\).
Hide middle: yes
Prefix Greater: the upper sum according to \((P,T)\) is
Suffix Greater: than the one according to \((P',T')\)

Smaller0: Given \(\epsilon>0\), there are functions \(g,h\in\mathcal R\) such that \(g\le f\le h\) and [$]\int_a^bh(x)-g(x)\dd x\le\epsilon[/$]
Greater0: \(f:[a,b]\to\mathbb R\) is Riemann integrable
Page: 173
Chapter: 3
Section: 2
Variables: \(a<b\) two reals
\(f:[a,b]\to\mathbb R\).
Index: 22
Hide middle: yes
Name: Sandwich principle

Name: Zeno's staircase function
Definition: \(f(x)=1-\frac{1}{2^{n} }\) for \(x\in\left[1-\frac{1}{2^{n-1} },1-\frac{1}{2^{n} }\right)\).
Page: 174
Chapter: 3
Section: 2
Kind: Chap

Smaller0: Riemann integrable
Greater0: bounded
Page: 175, 167
Chapter: 3
Section: 2
Variables: \(f:[a,b]\to\mathbb R\).
Index: 20, 23
Kind: Theorem
Smaller to greater: iff
Prefix Greater: \(f\) is
Greater1: set of discontinuity points is a zero set
Greatest: \(f\) is Darboux integrable

Smaller0: \(f\) is integrable and [$]\int_a^bf(x)\dd x=0[/$]
Greater0: \(f(x)=0\) at every continuity point \(x\) of \(f\)
Page: 181
Chapter: 3
Section: 2
Variables: \(a<b\) two reals,
\(M>0\) real,
\(f:[a,b]\to\mathbb R\).
Index: 31
Kind: Cor
Hide middle: yes
Smaller1: \(f\ge0\)
Extra: multiple condition

Smaller0: \(f,g\) continuous on \([a,b]\) and differentiable on \((a,b)\)
Greater0: there is a \(\theta\in(a,b)\) such that \(\Delta fg'(\theta)=\Delta g.f'(\theta)\)
Page: 152
Chapter: 3
Section: 1
Variables: \(f,g:[a,b]\to\mathbb R\).
Index: 6
Kind: Theo
Hide middle: yes
Name: Ratio mean value theorem

Name: The oscillation of \(f\) at \(x\)
Definition: \[\lim \sup_{t\to x}f(t)-\lim\inf_{t\to x}f(t)\]
Page: 177
Chapter: 3
Section: 2
Variables: \(a<x<b\) reals,
\(f:[a,b]\to\mathbb R\).
Definition2: \[\lim_{r\to 0,r\ge0}\text{diam}f[x-r,x+r].\]
Abbreviation: \(\text{osc}_x(f)\)

Name: Riemann-Lebesgue Theorem
Definition: \(f\) is Riemann integrable iff \(f\) is bounded and its set of discontinuity points is a zero set
Page: 175
Chapter: 3
Section: 2
Index: 23
Kind: Theo
Extra: \(f\) is Riemann integrable

Definition: the set of function \(f\) such that \(f\) is \(r\)th order differentiable and \(f^{(r)}(x)\) is continuous.
Page: 157
Chapter: 3
Section: 1
Notation: \(C^r\)

Smaller0: \(f\) is continuous on \([a,b]\) and \(f\) is differentiable on the interval \((a,b)\)
Greater0: there exists a point \(\theta\in(a,b)\) such that \(f(b)-f(a)=f'(\theta)(b-a)\)
Page: 151
Chapter: 3
Section: 1
Variables: \(f:[a,b]\to\mathbb R\).
Kind: Theo
Hide middle: yes
Name: Mean Value Theorem

Question: [$]\int_0^1z(x)\dd x=[/$]
Answer: \(\frac23\)
Page: 174
Chapter: 3
Section: 2
Variables: \(z\) be Zeno's staircase function
Kind: Chap

Question: Example of a derivative with a non jump discontinuity
Answer: \[f(x)=\left\{ \begin{array}{lll}x^2 \sin\frac 1x&\text{if }& x>0\\0&\text{if }&x\le 0\\\end{array}\right.\]
whose derivative is
\[f'(x)=2x\sin \frac1x-\cos \frac 1x\] for \(x>0\) and \(f'(x)=0\) for \(x<0\).
Page: 156
Chapter: 3
Section: 1

Question: \(F'(x)\)
Answer: exists and equals \(f(x)\) at every point \(x\) at which \(f\) is continuous.
Page: 183
Chapter: 3
Section: 2
Variables: \(f:[a,b]\to \mathbb R\) Riemann integrable
\(F(x)=\int_a^x f(t)dt\)
Index: 34
Kind: Cor

Smaller0: \(\lim\sup_{k\to \infty}\sqrt[k]{|a_k|}<1\)
Greater0: \(\sum a_i\) converges
Page: 194
Chapter: 3
Variables: \(\sum a_i\) a series
Index: 42
Hide middle: yes
Name: Root test
Extra: Duplicate

Smaller0: \(\lim\sup_{k\to\infty}\left|\frac{a_{k+1}}{a_k}\right|<1\)
Greater0: converges
Page: 195
Chapter: 3
Variables: \(\sum a_i\) a series.
Index: 43
Hide middle: yes
Prefix Greater: the series
Name: Ratio test

Smaller0: \(f\) is continuous
Greater0: \(f\) has an antiderivative
Page: 185
Chapter: 3
Section: 2
Variables: \(a<b\) two reals,
\(f:[a,b]\to\mathbb R\).
Index: 36
Kind: Corollary

Smaller0: \(\int_a^b \left(f_0(x)f_{1}'(x)+f_0(x)'f_{1}(x) \right)\mathtt{d} x\)
Greater0: \(\left[f_0(x)f_1(x)\right]_{a}^b\)
Page: 189
Chapter: 3
Section: 2
Variables: \(f_0,f_1:[a,b]\to \mathbb R\).
Index: 39
Smaller to greater: =
Hide middle: yes
Assuming: \(f_0',f_1'\in\mathcal R\)
Name: Integration by Parts

Name: The improper Riemann integral of \(f\)
Definition: [$$]\lim_{c\to b}\int_a^c f(x)\dd x[/$$].
Page: 190
Chapter: 3
Section: 2
Variables: \(f:[a,b)\to\mathcal R\).
Notation: [$]\int_a^bf(x)\dd x[/$]

Smaller0: \(\sum a_k\) converges
Greater0: \(\forall\epsilon>0\exists N\) such that \(m,n\ge N\implies{\left| \sum_{k=m}^na_k\right|<\epsilon}\)
Page: 191
Chapter: 3
Variables: \(\sum a_i\) a series.
Smaller to greater: iff
Hide middle: yes
Name: Cauchy Convergence Criterion for series

Smaller0: \(\lim\sup_{k\to \infty}\sqrt[k]{|a_k|}>1\)
Greater0: \(\sum a_i\) diverges
Page: 194
Chapter: 3
Variables: \(\sum a_i\) a series.
Index: 42
Hide middle: yes
Name: Root test
Extra: duplicate
Context: Series

Smaller0: \(\lim\sup_{k\to\infty}|\frac{a_{k+1}}{a_k}|>1\)
Greater0: diverges
Page: 195
Chapter: 3
Variables: \(\sum a_i\) a series.
Index: 43
Hide middle: yes
Prefix Greater: the series
Name: Ratio test

Smaller0: \(g\) continuously differentiable bijection
Greater0: [$]\int_a^b f(y)\dd y[/$]
Page: 189
Chapter: 3
Section: 2
Variables: \(f:[a,b]\to\mathbb R\) Riemann integrable,
\(g:[c,d]\to[a,b]\).
Index: 38
Hide middle: yes
Smaller1: \(g'>0\)
Name: Integration by substitution
Greater1: [$]\int_c^df(g(x))\dd g(x)=\int_c^d (f\circ g)(x)g'(x)\dd x[/$]
Connect Greater0: equals

Name: Discontinuous function with an antiderivative
Chapter: 3
Section: 2
Kind: Cor
Example: \(f(x)\) is \(0\) if \(x\le 0\) and \(\sin\left(\frac\pi x\right)\) otherwise
Example2: \(\mathcal X_{\mathbb R^{>0}}\).

Definition: \(\lim_{\lambda(P)\to 0}s(f;P)\)
Page: 341
Chapter: 6
Section: 1
Variables: \(n\in\mathbb N\),
\(I\subseteq\mathbb R^n\) an interval
\(f:I\to\mathbb R\)
Notation: \(\underline I\)
Index: 3
Kind: Prop

Name: \(\mathcal R[a,b]\)
Page: 342, 347
Section: 1.3 c, d
Chapter: 6
Index: 26, 28, 29, 30 (Real pugh)
4, 3 (Zorich)
Kind: Prop, Ex
Extra: Zorich P. 342, 347, Ch 6 1.3c,d
Real, pugh. P 180, Ch3.2, Ch 4.1 theo 6
Under: pointwise sum
Condition: finite
CounterExample: Let \((q_i)_i\) an enumeration of the rationals.
\(\mathcal X_{\{q_i\}}\) is not Riemann integrable
Under2: pointwise product
Under3: norm
Under4: restriction to an interval
Under5: composition on the left by a function
Condition5: continuous, not arbitrary
CounterExample5: \(\mathcal X_{\mathbb R\setminus\{0\}}\circ\text{Rational ruler}=\mathcal X_{\mathbb Q}\)
Not6: not
Under6: composion
Under7: product by a scalar
Under8: adding a scalar
Under9: Uniform limit
Other fields (not shown on cards):
Prefix7: \(\mathcal X_{\mathbb Q}=\mathcal X_{\mathbb R\setminus\{0\}}\)

Smaller0: \(\lim_{x\to a+0}g(x)=\infty\)
Greater0: \(\lim_{x\to a+0}\frac{f(x)}{g(x)}=\lim_{x\to a+0}\frac{f'(x)}{g'(x)}\)
Page: 248
Chapter: 5
Section: 4.4
Variables: \(-\infty\le a<b\le \infty\)
\(f,g:]a,b[\to\mathbb R\)
Index: 8
Kind: Prop
Assuming: \(g'(x)\ne 0\) for \(x\in]a,b[\)
both functions are differentiable
Smaller1: (\(\lim_{x\to a+0}f(x)=0\) and \(\lim_{x\to a+0}g(x)=0\))
Name: l'Hôpital's rule
Extra: No condition for \(\lim_{x\to a+0}f(x)\) in the infinity case
Zorich Chap 5, 4.4, page 248
Pugh, Real analysis, Ch 3, 1, page 153
Connect Smaller0: or

Name: The set of Riemann integrable function
Page: 182
Section: 2
Chapter: 3
Index: 32
Kind: Cor
Under: right-composition by \(C^1\)-diffeomorphism
Under2: right-composition by a homeomorphism whose inverse satisfies a Lipschitz condition
Under3: sum, product...

Smaller0: \(f\) is \((r+1)\)th order differentiable at \(x\)
Greater0: \(\frac{f^{r+1}(\theta)}{(r+1)!}h^{r+1}\) for some \(\theta\in(x,x+h)\)
Page: 160
Chapter: 3
Section: 1
Variables: \(a<x<b\in\mathbb R\)
\(f:(a,b)\to\mathbb R\)
Index: 12
Prefix Greater: \(R(h)\) is of the form

Question: Tools to check whether a series \(\sum_ia_i\) absolutely converge
Answer: Root test: \(\lim\sup_{k\to \infty}\sqrt[k]{|a_k|}<1\)
Variables: \((a_i)_{i\in\mathbb N}\)
Context: \(a_i\)
Answer2: \(|a_i|<b_i\) with \(\sum b_i\) converging (or similarly \(|a_{\lfloor x\rfloor}|<f(x)\) and \(\int_{0}^\infty f dx\) converging), appart for a finite number of \(i\)
Answer3: \(\lim\sup_{k\to\infty}\left|\frac{a_{k+1}}{a_k}\right|<1\)
Answer4: Cauchy: \(\forall\epsilon>0\exists N\) such that \(m,n\ge N\implies{\left| \sum_{k=m}^na_k\right|<\epsilon}\)
Answer5: \(\sum2^ka_{2^k}\) converges with \(a_k\) decreasing positive