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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7
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2answers
52 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
2answers
681 views

Construct an example of a 4×4 matrix, with one of its eigenvalues equal to −3, that is not diagonal or invertible, but is diagonalizable

Construct an example of a 4×4 matrix, with one of its eigenvalues equal to −3, that is not diagonal or invertible, but is diagonalizable. I know how to find the eigenvalues, and diagonalizing ...
2
votes
1answer
15 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 ...
4
votes
1answer
262 views

Mid-point convexity does not imply convexity [duplicate]

A function $f: X \rightarrow \mathbb{R}$ is said to be mid-point convex if for all $x, y \in X$, we have $$f(\frac{x + y}{2}) \leq \frac{f(x) + f(y)}{2}. $$ Can you please give an example of a ...
2
votes
2answers
37 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
17 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
votes
1answer
61 views

Counterexample to proposition on $\sigma$-algebras

If $\sigma$-algebras $\mathscr{F}$ and $\mathscr{G}$ are independent and if sigma-algebras $\mathscr{F}$ and $\mathscr{H}$ are independent then $\mathscr{F}$ and $\mathscr I := ...
0
votes
2answers
29 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
36 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
4answers
76 views

Counterexample to “$A \to B, A \to C$, therefore $B \to C$”

We have $A\to B$ and $A\to C$. I need counter-examples to: '$\therefore B\to C$'. More formally, disprove: $$ (A\to B)\land(A\to C)\to (B\to C)$$ I have $A$ is a blackbird, $B$ is 'is black', $C$ ...
1
vote
1answer
46 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 ...
0
votes
1answer
1k views

Proving/Disproving set identity $(A\cap B)\cup C= A\cap (B\cup C)$ [duplicate]

For any sets $A$, $B$, $C$; $(A\cap B)\cup C= A\cap (B\cup C)$ I understand that this means that (A and B) or C = A and (B or C), but how would you prove or disprove these set identities. Any ...
1
vote
1answer
62 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 ...
1
vote
0answers
40 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. ...
7
votes
6answers
557 views

An Example of a Nested Decreasing Sequence of Bounded Closed Sets with Empty Intersection

Could someone provide me with an example of a metric space having a nested decreasing sequence of bounded closed sets with empty intersection? I first thought of Cantor set but the intersection is not ...
0
votes
1answer
17 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 ...
0
votes
1answer
39 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 ...
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 ...
0
votes
0answers
35 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 ...
1
vote
2answers
25 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 ...
7
votes
2answers
427 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
28 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 ...
2
votes
8answers
119 views

Example where $x^2 = e$ has more than two solutions in a group [closed]

Show by means of an example that it is possible for the quadratic equation $x^2 = e$ to have more than two solutions in some group $G$ with identity $e$.
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vote
4answers
323 views

Example of a group in which the equation $x^2=e$ has more than two solutions

I am looking for an example of a group in which the equation $x^2=e$ has more than two solutions, where $e$ is the identity element. Groups with two solutions are easy to find: nonzero reals under ...
2
votes
1answer
41 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, ...
0
votes
2answers
60 views

Is the functions $f+g$ one to one?

I am reviewing for a test and there is this question. Functions $f: \mathbb R\to \mathbb R$, $g: \mathbb R\to\mathbb R$ are both one to one on the set of real numbers $\mathbb R$. Is the function ...
1
vote
2answers
63 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
34 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 ...
11
votes
2answers
425 views

Counterexample to “Measurable in each variable separately implies measurable”

Some fellow classmates are preparing for a qualifying exam on real analysis, and asked me for help on the following question: Let $ \ f:[0,1]^2\longrightarrow\mathbb{R}$ be such that: (i) $\ ...
1
vote
1answer
23 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
181 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. ...
2
votes
2answers
57 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 ...
1
vote
0answers
29 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 $x_0$ ?
1
vote
1answer
24 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 ...
1
vote
1answer
108 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 ...
2
votes
2answers
2k views

Example of a general random variable with finite mean but infinite variance

Given a probability triple $(\Omega, \mathcal{F}, \mu)$ of Lebesgue measure $[0,1]$, find a random variable $X : \Omega \to \mathbb{R}$ such that the expected value $E(X)$ converges to a finite, ...
3
votes
3answers
756 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, ...
0
votes
1answer
35 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 ...
3
votes
4answers
3k views

iid variables, do they need to have the same mean and variance?

If two random variables $x$ and $y$ are identical and independently distributed, do they need to have the same mean and variance? Can there exist a case where they are iid and still have different ...
1
vote
1answer
63 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
44 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
73 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
23 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 $2^{n^2}>c\cdot 2^n$?
2
votes
2answers
44 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 ...
1
vote
2answers
33 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 ...
9
votes
5answers
3k views

Examples for subspace of a normal space which is not normal

Are there any simple examples of subspaces of a normal space which are not normal? I know closed subspace of a normal space is normal, but open subspace in most cases which I can think of are ...
0
votes
1answer
37 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
50 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 ...
1
vote
2answers
65 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