Examples and counterexamples are great ways to learn about the intricacies of definitions in mathematics. Counterexamples are especially useful in topology and analysis where most things are fairly intuitive, but every now and then one may run into borderline cases where the naive intuition may ...

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8
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1answer
314 views

Continuous functions and uncountable intersections with the x-axis

Let $f : \mathbb{R} \to \mathbb{R}$ such that the set $X = \{x \in \mathbb{R} : f(x) = 0\}$ does not contain any interval (i.e. there is no interval $I \subset X$) Of course the set $X$ can be ...
2
votes
0answers
108 views

Game theory, Book by Tirole and Fudenberg, Never a weak best response,unclear example

In this book, I have the following problem: on page 446, there is a sentence: Note that $(0.9,0.9)$ is not removed by NWBR, as D is not dominated after C is deleted. I do not understand this "as". ...
2
votes
1answer
40 views

Large deviation theory--examples of irregular sets

Let $(X,\tau)$ be a topological space, let $\mathcal{B}$ be its Borel $\sigma$-algebra, and $\mu_\epsilon$ be a family of probability measures on $(X,\mathcal{B})$. Suppose also that $\mu_\epsilon$ ...
1
vote
3answers
76 views

$f'$ changes strict monotonicity but $f''$ isn't of strictly opposite signs

Let $I \subset \mathbb{R}$ be an open interval and $f \in C^2(I,\mathbb{R})$. I am looking for an (simple) example of $f$ with the following properties ($x_0 \in I$) $f'$ is strictly monotonic ...
1
vote
2answers
103 views

Is there a well known nontrivial counterexample to this claim?

Suppose we have $A\subseteq\mathbb{N}$ with the property that if $B\subseteq\mathbb{N}$ and $B$ is finite, then $\exists a\in A\setminus\{ 1\}$, $\forall b\in B$, $\gcd(a,b)=1$. Are there any well ...
2
votes
1answer
54 views

Are there functions that are Holder continuous but whose variation is unbounded?

I have recently been introduced to the concept of Holder condition and I was told that there are functions that are Holder continuous but whose variation in unbounded. Can anyone present an example, ...
7
votes
2answers
53 views

Non-measurable sets on $\mathbb{N}$

I'm familiar with the "construction" of non-measurable sets on $\mathbb{R}$. But of interest to me is if there is a way to construct a countably additive probability measure $\mu$ on $\mathbb{N}$ such ...
2
votes
1answer
17 views

Eigenvalue alteration counter-proof

If $A$ is a square $n\times n$ matrix, with $\lambda_1,\ldots,\lambda_n$ being the eigenvalues of $A$, $v_1$ being the eigenvector associated with eigenvalue $\lambda_1$, and $d$ the column vector of ...
0
votes
1answer
24 views

Newton's Method in unconstrained optimization fails to converge

In order to show that Newton's method can produce a sequence of iterates that diverges, an example given in my book is apply Newton's Method to minimize $f(x)={2\over 3}|x|^{3\over 2}$. starting at ...
2
votes
2answers
55 views

Constructing a (smooth) diffeomorphism between non-smooth manifolds

I've been trying to construct a smooth diffeomorphism between non-smooth manifolds. Unfortunately I don't think I know enough manifolds well enough to find an example of this. Mostly I've been ...
0
votes
1answer
43 views

How do I prove the Poisson integral formula for a harmonic function on exterior of a disk?

Let $R>0$ and $u:\mathbb{C}\setminus B(0,R)\rightarrow \mathbb{R}$ be a continuous function such that $u$ is harmonic on $\mathbb{C}\setminus\overline{B(0,R)}$. Assume that $u$ is bounded at ...
0
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2answers
31 views

Question Regarding the Commutativity of F-Algebras when the Algebra is finite dimensional over F.

Let $A$ be some $F$-Algebra, for some field $F$, with the property that $A$ is finite dimensional over $F$. Is $A$ always commutative?
0
votes
1answer
44 views

Counter example for - product of general cardinal separable spaces

I am looking for a counter example for the claim that a product (of any cardinal) of separable spaces is separable, I saw in Uncountable product of separable spaces is separable? and On the ...
1
vote
1answer
51 views

Finding a continuous $f\in C(\mathbb{T})$ with $\{\hat{f}(n)\}\notin \ell^p$

Denote the circle by $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Find an example for a continuous function $f\in C(\mathbb{T})$ with coefficients of the fourier series, $\{\hat{f}(n)\}\notin\ell^p,\forall ...
0
votes
1answer
33 views

What is an example of such series?

Related:Why is this sequence uniformly convergent? Let $K$ be a compact subset of $\mathbb{C}$. Let $f_n$ be a sequence of continuous functions such that $f_n:K\rightarrow \mathbb{C}$. Assume that ...
2
votes
2answers
52 views

Normed algebra with bounded multiplication

Sometimes one finds the following definition of a normed algebra: This is an algebra with a norm on the underlying vector space such that there is a constant $K \geq 0$ such that $|x \cdot y| \leq K ...
1
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1answer
63 views

Category theory: do other examples of “resplendent” properties exist?

Call a predicate $P$ defined on categories resplendent iff it satisfies the following condition: for all categories $\mathbf{D}$, if $P(\mathbf{D}),$ then for all categories $\mathbf{C}$, we have ...
2
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0answers
42 views

What's the name of this extremely common but extremely pathological continuous function?

Okay, so let's define a random function $F$, such that the value of $F(x)$ is uniformly distributed on $[-1,1]$, and such that for any $x$ and $y$ with $x \ne y$, $F(x)$ and $F(y)$ are independent. ...
0
votes
1answer
20 views

What would be a non-injective holomorphic function on $B(0,\epsilon)$?

Let $\epsilon > 0$. What would be a non-injective holomorphic function on $B(0,\epsilon)$ such that $f'(0)\neq 0$? Since $f'(0)\neq 0$, there exists a neighborhood of $0$ such that $f$ is ...
2
votes
1answer
21 views

Determining if a set is measurable by upper and lower sets

I have the following question regarding Lebesgue measure: If $A,B$ are measurable sets and I have $m(A\setminus E)=0$ and $m(E\setminus B)=0$, is it enough to determine that $E$ is measurable? We do ...
0
votes
1answer
42 views

If $\int_a^x f(t)\, dt$ is differentiable, is its derivative integrable?

Let $f$ be a real-valued Lebesgue integrable function on $[a,b]$. If $F(x) = \int_a^x f(t)\, dt$ is differentiable on $[a,b]$, is $F'(x)$ (Lebesgue) integrable? I know there are examples of ...
0
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0answers
36 views

There is a function with this condition…

Show that there is a function $f : P(\Bbb{N} ) \to \Bbb{N}$ ,( $P(\Bbb{N})$ is power set) with this condition that for $A \in P(\Bbb{N})- \{\emptyset \}$ , $f(A) \in A$ , is there function with this ...
2
votes
2answers
96 views

If $|a^{2}|=|b^{2}|$ then $|a|=|b|?$

If $|a^{2}|=|b^{2}|$ (for non identity elements $a$ and $b$ of a group $G$ and $|a|$ denotes the order of the element $a$) prove or disprove that $|a|=|b|.$ I tried as follows Clearly infinite order ...
1
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2answers
40 views

Example of an infinite non-normal subgroup $H$ of non-abelian (topological)group $G$.

I am trying to find Example of an infinite non-normal subgroup $H$ of non-abelian (topological)group $G$. For instance, The subset $\mathrm{T}(n,\mathbb{R})\subset \mathrm{GL}(n, \mathbb{R})$ of ...
1
vote
1answer
34 views

Uniform convergence does not closed under multiplication

Construct sequences $\{f_n\}$, $\{g_n\}$ which converge uniformly in some set $E$, but such that $\{f_ng_n\}$ does not converge uniformly on $E$ (of course, $\{f_ng_n\}$ must converge on $E$). My ...
7
votes
2answers
455 views

Is a continuous function locally uniformly continuous?

Assume a function, $f : X \to Y$, mapping between two metric spaces, $X,Y$, is pointwise continuous, i.e. for every $\varepsilon >0$ and $x \in X$ there exists a $\delta>0$ such that $$ ...
2
votes
1answer
53 views

Example of quasi-compact, non-quasi seperated scheme where qcqs fails?

The qcqs lemma (in Ravi Vakil's notes) says that if $X$ is a quasi-compact (qc) and quasi-separated (qs) scheme, for any global section $f$, the natural map from $\Gamma(X, O_X)_f \to \Gamma(X_f, ...
1
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2answers
69 views

Are supersets of non-empty measurable sets measurable?

Challenging conventional wisdom question Let $(S, \Sigma, \mu) = (\mathbb{R}, \mathscr{M}(\mathbb{R}), \lambda)$, where $\mathscr{M}(\mathbb{R})$ is the collection of all $\lambda$-measurable subsets ...
0
votes
1answer
38 views

Are supersets of measurable sets measurable?

Challenging conventional wisdom question Let $(S, \Sigma, \mu) = (\mathbb{R}, \mathscr{M}(\mathbb{R}), \lambda)$, where $\mathscr{M}(\mathbb{R})$ is the collection of all $\lambda$-measurable subsets ...
1
vote
1answer
26 views

finding maximum matching of a graph from an optimal proper coloring of complement of graph

Let $G:=(V,E)$ be a simple undirected graph. Let $\bar{G}$ denote the complement of $G$. Let $c:V\rightarrow \{1,2,...,\chi(\bar{G})\}$ be a proper coloring of $\bar{G}$. It is clear that the sets of ...
1
vote
1answer
242 views

Which of the following statements are true on countable sets

Show that the numbers of the form $\sum_{k=1}^{\infty} \frac{a_j}{3^k}$ , where $a_j = 0$ or $a_j = 1$ is countable . If $A = \cap_i^n A_1$ is countably infinite, then atleat one $A_i$ is counntable. ...
1
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0answers
37 views

Problems with continuous and derivable functions

If a function $\mathbb{R} \to \mathbb{R}$ is continuous or derivable at a point $x_0$ of its domain, is also continuous or derivable in a neighbourhood of $x_0$ ?
2
votes
1answer
64 views

Examples where $R/I\cong R$? [duplicate]

I had to prove on a test that if $R$ is a PID then every surjective endomorphism of $R$ is an injection. To do this, I supposed there was a surjective endomorphism $\varphi:R\to R$. Then ...
1
vote
1answer
117 views

Humorous mathematical essays

Even though there are plenty examples of mathematical jokes, the mathematical literature is (in many cases) pretty dull. Nevertheless, examples exist in which an essay makes you smile with a nice pun ...
1
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1answer
28 views

Show the existence of a function from given sequence related to supremum.

I'm considering the following problem: Let $f(x)=\limsup_{\varepsilon\to0}\{f(y)\mid |x-y|\le\varepsilon\}$ for a upper semicontinuous function $f:(0,1)\to\mathbb{R}$. Assume that a sequence ...
0
votes
1answer
40 views

Example of two integrable functions/random variables whose product is not integrable?

If two functions/random variables are integrable and independent, then their product is integrable. What if they are not independent? What is an example? What I tried: Let $X, Y \in \mathscr ...
1
vote
1answer
32 views

Show that $f(x) = \log\log(\frac{e}{\Vert x \Vert_{2}})$ is in Sobolev space $W^{1,p}(B_1(0))$

As part of my seminar this semester, I need to show that $f(x) = \log\log(\frac{e}{\Vert x \Vert_{2}}) \in W^{1,p}(B_1(0))$. I have shown that $f$ is indeed in $L^p$, but could use some help proving ...
3
votes
3answers
770 views

$f$ bounded but $f'$ isn't

Is there a bounded function $f$ that holomorphic on the open unit disc but $f'$ isn't bounded? I think first $f$ shouldn't be analytic outside the unit disc then we can't use Cauchy's inequality, ...
1
vote
1answer
65 views

Define a bijection [duplicate]

Okay so I know that I asked this already but I want to ask how much progress I have made. so the question is Construct a bijection between $[1,2]$ and $[3,5)$ So I have: \begin{equation} f(x) = ...
0
votes
1answer
50 views

A nowhere continuous function that maps compact sets to compact sets

Construct an example of a function $f:\mathbb{R}\to \mathbb{R}$ that is not continuous at any point, but satisfies the property "$f(K)$ is compact, when $K$ is compact" however $f(\mathbb{R})$ is ...
0
votes
3answers
77 views

A $\implies$ ¬A. Is my reasoning on the following example correct?

I already asked the question in the Philosophy forum, but I haven't gotten any answer, yet. Maybe the mathematicians will be faster: Oh, you// anything that rhymes is not true. If this ...
1
vote
1answer
24 views

Disproving big $O$ identity

How can I disprove that $2^{(n^2)}=O(2^n)$? Should I show that $\forall c >0$ we have $2^{n^2}>c\cdot 2^n$?
1
vote
2answers
39 views

Let $ f,g : A \rightarrow A (A \subset R)$ be two bijections. Give examples that f+g and fg are not bijections

Let $ f,g : A \rightarrow A (A \subset R)$ be two bijections. Give examples that f+g and fg are not bijections. I am not sure how to approach this question. I try to think of examples, but I ...
0
votes
1answer
41 views

If $f:[0,\infty) \to \mathbb{R}$ is continuous and $f(x) > 0$ for all $x$, then there exists $c > 0$ such that $f(x)\geq c$ for all $x$.

If $f:[0,\infty) \to \mathbb{R}$ is continuous and $f(x) > 0$ for all $x$, then there exists $c > 0$ such that $f(x)\geq c$ for all $x$. I feel that this is a false statement given the order of ...
0
votes
1answer
58 views

If $f:[0,1) \to \mathbb{R}$ is continuous and $f(x) > 0$ for all $x$, then there exists $c > 0$ such that $f(x)\geq c$ for all $x$.

If $f:[0,1) \to \mathbb{R}$ is continuous and $f(x) > 0$ for all $x$, then there exists $c > 0$ such that $f(x)\geq c$ for all $x$. I feel that this is a false statement given the order of the ...
2
votes
2answers
68 views

What would be a counterexample to Cauchy's integral formula or Cauchy's theorem?

Here is the Cauchy's theorem. Let $G$ be open in $\mathbb{C}$. (Not necessarily connected) Let $f:G\rightarrow \mathbb{C}$ be a holomorphic function. Let $\gamma_k$ be closed rectifiable ...
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vote
2answers
70 views

Can any real number be expressed as the sum of a rational number with a number of the Cantor set?

I only ask for either a proof that this is true or a counterexample of a real number that can NOT be expressed as the sum of a rational + a number in the Cantor Set Thanks
2
votes
2answers
59 views

A Markov process which is not strong Markov process (follow up 2)

In http://mathoverflow.net/questions/43833/a-markov-process-which-is-not-a-strong-markov-process George Lowther's example: "Consider the following continuous Markov process $X$, starting from ...
3
votes
1answer
49 views

Counterexamples of Annihilator in Infinite Dimensional Setting

Is there a simple example of an infinite dimensional vector space such that $(W_1\cap W_2)^\circ \not= W_1^\circ +W_2^\circ$?
2
votes
3answers
225 views

Composition of nonlinear maps

Let: $$f: V \to W$$ and $$g: W \to U$$ where V, W, and U are some vector spaces. Can you please give an example (an interesting one, if possible) where $$g \circ f$$ is a homomorphism where f or g are ...