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 ...

learn more… | top users | synonyms (2)

0
votes
0answers
30 views

Normal extension of a field

Let $F$ be an extension of $K$ (they are both fields). I know that if $F$ has finite degree over $K$, then the following things are equivalent: 1) $F$ is such that every irreducible polynomial in ...
6
votes
1answer
81 views

Examples of provably${}^n$ unprovable statements

Given any statement $A$ and a classical theory $T$ which we assume is at least as strong as Peano Arithmetic ($\sf PA$), we have that $T\vdash A$ implies $T\vdash T\vdash A$ (that is, if a statement ...
3
votes
1answer
31 views

What is an example of a uniformly continuous function but not absolutely continuous

Is there a function that is uniformly continuous function but not absolutely continuous. My answer is $f(x)=x^{2}, \forall x\in R$ Is this right? Are there any other?
0
votes
1answer
58 views

Continuous or Differentiable but Nowhere Lipschitz Continuous Function

What is a real valued function that is continuous on a close interval but not Lipschitz continuous on any subinterval? What is a real valued function that is differentiable on a close interval but ...
0
votes
0answers
75 views

Prove or disprove $2abc(a+b+c)\ge 3(a^2b^2c^2+1)$

Let $a,b,c>0,ab+bc+ca=3$, prove or disprove $$2abc(a+b+c)\ge 3(a^2b^2c^2+1)$$ Now I can't find any counterexample
1
vote
1answer
41 views

Can you give me an example of a function that is either upper OR lower quasi-continuous but not both?

A function $f: X \rightarrow \mathbb{R}$ is said to be upper (lower) quasi-continuous at $x \in X$ if for each $\epsilon >0$ and for each neighbourhood $U$ of $x$ there is a non-empty open set $G ...
4
votes
0answers
25 views

What are types of coalgebras that are more naturally described by cooperads?

Let $\mathsf{C}$ be a symmetric monoidal category. An object $X \in \mathsf{C}$ has two operads "naturally" (the two constructions aren't functorial) associated to it: the operad of endomorphisms and ...
7
votes
4answers
494 views

Non-separable compact space

Off the top of my head, I can't think of a non-separable compact space. Can you provide a good example?
1
vote
1answer
32 views

A graph with list chromatic number $4$ and chromatic number $3$

What is an example of a graph with chromatic number $\chi(G)=3$ and list-chromatic number $\chi_\ell(G)=4$? My first thought was to consider complete tripartite graphs since these will have chromatic ...
0
votes
1answer
24 views

Torsion elements and subgroups of nonabelian group

I am currently studying torsion groups and I am playing around with defintions to get used to them. An element $g \in G$ is a torsion element, if there exists $n \in \mathbb{N}$ so that $g^n = e$, ...
0
votes
1answer
16 views

Example of an uncountable metric space where every point is isolated

I was trying to come up with an example of an uncountable metric space all of whose points are isolated. I've had difficulty thinking of one, has anyone got any nice examples? Just in case: ...
0
votes
1answer
36 views

Question on abstract algebra about invertiblity?

Let $(G,*)$ be a group. Could anyone give me an example $a,b\in G$ such that $$a*b=e\mbox{ and } b*a\neq e $$ Where $e$ is the identity element. I would appreciate any help. Thanks in advance!
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, ...
12
votes
2answers
418 views
1
vote
1answer
67 views

Examples of the use of $(p \vee q) \wedge (\neg p \vee r) \Rightarrow (q \vee r)$ in “real” mathematics?

Here, a proof is requested for the following tautology: $$(p \vee q) \wedge (\neg p \vee r) \Rightarrow (q \vee r)$$ Its pretty easy to prove; nonetheless, I don't find the formula at all intuitive, ...
1
vote
1answer
37 views

Counterexample that a measurable function does not guarantee almost sure convergence

I try to find a counterexample that if $X_n\xrightarrow{\text{a.s.}}X$ then for a measurable function $g:\mathbb{R}\to \mathbb{R}$ this does not imply that $$g(X_n)\xrightarrow{\text{a.s.}}g(X)$$ ...
1
vote
0answers
35 views

Ergodic actions of orthogonal group $O(2)$

I am looking for explicit ergodic action of $O(2)$ on a von Neumann algebra $M$. ($O(2)$=orthogonal group of $2\times 2$ matrix)
2
votes
1answer
33 views

Is every such family induced by an associative operation?

Suppose we're given an associative operation $\star : X \times X \rightarrow X$. Then for each $n \in \mathbb{N}_{>0}$, there's a function $f_n : X^n \rightarrow X$ given as follows: ...
9
votes
4answers
951 views

Does a continuous point-wise limit imply uniform convergence?

Question Given a sequence of continuous functions $(f_n)_{n \in \mathbb N}$ and define $$ f : X \rightarrow Y, \quad f(x) = \lim_{n \rightarrow \infty} f_n(x) $$ where $X$ and $Y$ are metric spaces. ...
2
votes
1answer
111 views

inflexion points of a composition of functions

Let's consider a smooth positive bounded and non-increasing function $h$ over $\mathbb{R}^{+}$ (for example some kind of decreasing sigmoid). A) Is it true that if $h$ has only one inflexion point, ...
-1
votes
2answers
25 views

Example for associative, commutative operations

I need examples of binary operations for real numbers that are associative and commutative associative but not commutative The examples are for a programming class and need to be rather simple. ...
1
vote
0answers
61 views

Counterexample for “Filtration of stopping time equals filtration generated by stopped process”

I am working in a discrete setting. Consider any stochastic process $(X_n)_{n\in\mathbb N}$ with its natural filtration $(\mathcal F_n)_{n\in\mathbb N}$ and a stopping time $\tau$. We know that ...
6
votes
1answer
72 views

A smooth nowhere analytic function such that all derivatives are monotone

Related questions that might provide some context: (1) (2) (3) (4) Let's restrict our attention to real-values functions on an open unit interval $f:(0,1)\to\mathbb R$. There are examples ...
3
votes
1answer
32 views

How many examples exist of Lie groups that are 2-dimesional surfaces?

It is relatively easy to show that $\mathbb{R}^2$ or $\mathbb{T}^2$ are 2-dimensional surfaces with a structure of Lie groups. I can not find other surface which are also a Lie group, there are more ...
4
votes
1answer
37 views

Sum of closed spaces is not closed

I recently encountered the theorem that the sum of a closed (linear) subspace with a finite dimensional subspace is closed subspace of the Banach space in which it is contained. However, this came ...
1
vote
1answer
48 views

Is every set of measure zero countable?

I know it is true that every countable set has measure zero, but is the converse true. Is it true that every set of measure zero is countable?
13
votes
1answer
201 views

Find $f:C\to\mathbb{R}^2$ continuous and bijective but not open, $C\subset\mathbb{R} ^2$ is closed and connected

Are there a closed connected subspace $C$ of $\mathbb{R}^2$ and a continuous, bijective function $f:C\to\mathbb{R}^2$ that is not open? If we remove the condition for $C$ to be connected, we have ...
0
votes
1answer
8 views

Counter example for The Composition Theorem for Riemann Integrability

Based on The Composition Theorem (found in the last Lemma here) we can say that if a function f is Riemann Integrable, then $f^n$ is Riemann Integrable as well. The converse is not true, but I can't ...
6
votes
1answer
107 views
0
votes
3answers
30 views

Continuous for each variables does not implies continuous

Prove or disprove the following statement: Statement. Continuous for each variables, when other variables are fixed, implies continuous? More clearly, prove or disprove the following problem: Let ...
358
votes
34answers
22k views

Examples of apparent patterns that eventually fail

Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of 'proof'. I receive responses like: "surely if the Collatz ...
1
vote
0answers
17 views

Counterexamples of Cumulative Distribution Function ( multidimensional )

For simplicity, we consider 2-dimensional. We consider the function $F:\mathbb{R}^2\to\mathbb{R}$. Let $F$ satisfies: $0\leq F(x_1,x_2)\leq1$ for $(x_1,x_2)\in\mathbb{R}^2$ ...
1
vote
2answers
104 views

Non-subdifferentiable convex function on a normed space

Is there any convex function $f$ on a normed space $X$ such that $\partial f(x)=\emptyset$ at every $x\in X$?
9
votes
2answers
3k views

Give an example of a function $h$ that is discontinuous at every point of $[0,1]$, but with $|h|$ continuous on $[0,1]$

Give an example of a function $h:[0,1]\to\mathbb{R}$ that is discontinuous at every point of $[0,1]$, but such that the function $| h |$ that is continuous on $[0,1]$. I don't really even know where ...
0
votes
0answers
14 views

Baire class one function

in here http://www.m-hikari.com/ijma/ijma-2013/ijma-5-8-2013/feneciosIJMA5-8-2013.pdf has been shown that if a function $f$ is a real valued function on $\mathbb{R}$ with a countable set of ...
1
vote
1answer
35 views

Is there a graph with $n$ vertices and $n^2/4$ edges that isn't bipartite? [closed]

Is there a graph with $n$ vertices and $\lfloor n^2/4\rfloor$ edges that isn't bipartite and contains no triangles ($K_3$)? Rather, what I am asking is whether Mantel's Theorem implies that every ...
0
votes
2answers
24 views

$A^\circ \cup B^\circ \subset (A \cup B )^\circ$ Counterexample for = instead of $\subset$

If $A$ and $B$ are sets of real numbers, then $(A \cup B)^{\circ} \supseteq A^ {\circ}\cup B^{\circ}$ But the same relation with a = isn't always true. Can someone find an example where the = ...
1
vote
1answer
55 views

counterexample for $\overline{A \cap B} = \overline{A} \cap \overline{B}$

Prove $(\overline{A \cap B}) \subseteq \overline{A} \cap \overline{B}$. But the same relation with a $=$ isn't always true. Can someone find an example where the $=$ doesn't hold, I can't seem to ...
2
votes
2answers
63 views

Looking for an example of an infinite metric space $X$ such that there exist a continuous bijection $f: X \to X$ which is not a homeomorphism

I am looking for an example of an infinite metric space $X$ such that there exists a continuous bijection $f: X \to X$ which is not a homeomorphism. Please help. Thanks in advance.
7
votes
0answers
78 views

Algebra defined by $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$

Let $\cal A$ be the (noncommutative) unitary $\mathbb Z$-algebra defined by three generators $a,b,c$ and four relations $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$. Is it true that $ab\neq 0$ in $A$ ? This ...
0
votes
1answer
29 views

Is there an example where $ A \subseteq\mathcal {P}\bigcup A\ $ is no longer true?

I came up with the following: Let A = {x} Then $ \bigcup A = x $ $ \mathcal {P} \bigcup A =$ ? This is where I get stuck. The definition of power set is the set of all subsets of $A$ ...
7
votes
4answers
326 views

Ratio test for sequences, the other direction

Suppose I have a real sequence $x_n\to 0$. Is it true that: $$ \left|\frac{x_{n+1}}{x_n}\right|\to r<1 $$ for some $r\in\mathbb{R}$? If not, is it true that: $$\exists ...
0
votes
2answers
307 views

Distance from a compact subset need not be attained in a metric space?

Suppose we have a metric space $(X,d)$. Let $S$ be a compact subset of $X$. Provide me with an example of $X$, and $S$ (closed and bounded in $X$) such that $$\min \{d(p_0,p): p \in S\}$$ does not ...
0
votes
2answers
22 views

Example of a function $u \in W^{1,p}(\Omega)$ whose extension $\hat{u}$ to be $0$ outside $\Omega$ $\hat{u} \notin W^{1,p}(\mathbb{R}^n)$

Let's consider a function $u \in W^{1,p}(\Omega)$, where $W^{1,p}(\Omega)$ is the Sobolev Space and $\Omega$ is an open set. When we extend $u$ to $\hat{u}$ like this: ...
1
vote
2answers
51 views

example of a convergent series that $\lim \sup |\frac{z_{n+1}}{z_n} | > 1$

Let $(z_n) \subset \mathbb C$, with $z_n \neq 0$. It's known that if $\lim \sup |\frac{z_{n+1}}{z_n} | < 1$, so $\sum |z_n|$ converges, then $\sum z_n$ converges. Can I find an example of a ...
0
votes
0answers
20 views

An example that does not satisfy the conditions of the Fourier inversion theorem?

Here is the Fourier inversion theorem page in Wikipedia. It states that for every function $f(x)$ that satisfy some conditions ($f(x)$ can be a function such as a Schwartz function, an integrable ...
3
votes
1answer
46 views

Essential singularity of the resolvent operator of an unbounded operator

Is there an unbounded operator with isolated points in the spectrum, not all of which are eigenvalues? For unbounded operators it is known that isolated spectral points are either poles or essential ...
-4
votes
2answers
30 views

Series that converges on $[-1,1]$ [closed]

What is an example of a series that converges only on $[-1,1]$? I am unable to come up with one right now for some reason. Thanks
6
votes
2answers
87 views

Examples of Manifolds such that $\chi (X)=-3$

I am trying to come up with an example of a closed oriented manifold with euler characteristic equal to $-3$. I have tried to use $\chi (\underbrace{T^2\mathbin{\#}\cdots \mathbin{\#} T^2}_{\text{$g$ ...
5
votes
1answer
77 views

a dense set in plane

Is there a dense set in $\Bbb{R^2}$ that every vertical line or horizontal line intersect in finite points. I think that we can consider $\Bbb{Q} ×\Bbb{Q}$ but every vertical line or horizontal line ...