Title: Theory / Unimportant / Math / Analysis / Counter-examples=
Tags: questionExample

Question: Example of a field which can not be ordered.
Answer: \(\mathbb C\)
Page: 13
Chapter: 1

Question: A field which can be ordered in two distinct ways.
Answer: \(\mathbb Q[\sqrt 2]\), with positive set:
-the standard set of positive number, and
-the set of \(r-s\sqrt 2\) such that \(r+s\sqrt2\) is positive.
Page: 14
Chapter: 1
Section: 2

Question: An uncomplete ordered field
Answer: \(\mathbb Q\)
Page: 14
Chapter: 1
Section: 2

Question: A non-Archimedean ordered field
Answer: \(\mathbb R(x)\), with set of positive terms the polynomials \(\frac pq\) whose leading term has the same sign.
\(f/1\), for \(f\) non constant, is not bounded by any naturals.
Page: 15
Chapter: 1
Section: 4

Question: Example of ordered field that cannot be completed.
Answer: \(\mathbb R(x)\), with set of positive terms the polynomials \(\frac pq\) whose leading term has the same sign.
Page: 15
Chapter: 1
Section: 5

Question: Example of an ordered field where rationals are not dense.
Answer: \(\mathbb R(x)\), with set of positive terms the polynomials \(\frac pq\) whose leading term has the same sign.
Page: 15
Chapter: 1
Section: 6

Question: An ordered field which is Cauchy-complete but not complete
Answer: The quotient of Cauchy sequences of elements of \(\mathbb R(x)\), by the standard equivalence relation.
Page: 17
Chapter: 1
Section: 7

Question: Examples of integral domain without unique factorization
Answer: \(\mathbb Q[\sqrt 5]\): \(2\cdot 2=(1+\sqrt 5)(-1+\sqrt 5)\)
Page: 17
Chapter: 1
Section: 8

Question: An integral domain without GCD
Answer: \(\mathbb Q[\sqrt 5]\): \(4=(1+\sqrt 5)(-1+\sqrt 5)\) and \(1+\sqrt5\) hav no GCD
Page: 18
Chapter: 1
Section: 9

Question: Example of fraction that cannot be reduced to lowest terms uniquely
Answer: \(\mathbb Q[\sqrt 5]\):
\(\frac{2(1+\sqrt5)}{4}=\frac{1+\sqrt 5}2=\frac2{-1+\sqrt 5}\)
Page: 18
Chapter: 1
Section: 10

Question: Example of function continuous on a closed interval and not bounded there
Answer: [$]\defFun{\mathbb Q\cap[0,2]}{\overline{\mathbb Q}}{x}{\frac{1}{x^2-2}}[/$]
Page: 19
Chapter: 1
Section: 11
Index: a
Extra: todo: check in a paper version of the book

Question: Function continuous and bounded on a closed interval but not uniformly continuous there
Answer: [$]\defFun{\mathbb Q\cap[0,2]}{\mathbb Q}{x}{\substack {0\text{ if $0\le x< \sqrt2$}\\1\text{ otherwise}}}[/$]
Page: 19
Chapter: 1
Section: 11
Index: b

Question: Uniformly continuous function on a closed interval, not possesing a maximum value there.
Answer: [$]\defFun{\mathbb Q\cap[0,1]}{\mathbb Q}{x}{x-x^3}[/$]
Page: 19
Chapter: 1
Section: 11
Index: c

Question: Continuous function on a closed interval failing to have intermediate value property.
Answer: [$]\defFun{\mathbb Q\cap[1,2]}{\mathbb Q}{x}{x^2}[/$] fails to take value 2 or
Page: 19
Chapter: 1
Section: 11
Answer2: [$]\defFun{\mathbb Q\cap[0,2]}{\mathbb Q}{x}{\substack {0\text{ if $0\le x< \sqrt2$}\\1\text{ otherwise}}}[/$]
Index: d

Question: Nonconstant differentiable function whose derivate vanishes identically over a closed interval.
Answer: [$]\defFun{\mathbb Q\cap[0,2]}{\mathbb Q}{x}{\substack{0\text{ if $0\le x< \sqrt 2$}\\1\text{ otherwise}}}[/$]
Page: 19
Chapter: 1
Section: 11
Index: e

Question: differentiable function on which Rolle's theorem fails
Answer: [$]\defFun{\mathbb Q\cap[0,1]}{\mathbb Q}{x}{x-x^3}[/$]
Page: 19
Chapter: 1
Section: 11
Index: f

Question: A monotonic uniformly continuous nonconstant function having the intermediate value property, and whose derivative is identically 0 on an interval.
Answer: todo
Page: 19
Chapter: 1
Section: 11
Index: g