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)

3
votes
1answer
33 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
83 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 not ...
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 \...
6
votes
0answers
33 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
510 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
37 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
19 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!
12
votes
2answers
420 views
1
vote
1answer
68 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
40 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
63 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
34 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: $$f_n(x_0,\...
9
votes
4answers
1k 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
113 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, ...
6
votes
1answer
82 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 $\!^{[1]}$...
3
votes
1answer
37 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
43 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
49 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
202 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
12 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
108 views
0
votes
3answers
32 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 $\...
361
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
21 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 $(x_1,x_2)\in\mathbb{R}^2$ $\left\{\displaystyle\...
1
vote
2answers
105 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
18 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
36 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
25 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 = doesn'...
1
vote
1answer
58 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
69 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
81 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
33 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
328 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 N\in\mathbb{N}:\left|\frac{x_{...
0
votes
2answers
310 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
26 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: $$\hat{u}(x)=\left\{\begin{...
1
vote
2answers
52 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
22 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
52 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
8
votes
2answers
102 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
78 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 ...
2
votes
1answer
28 views

Integration obeys countable subadditivity?

Does Lebesgue integration have the property of countable subadditivity: 'if $f$ is integrable on $E$ and $E = \bigcup_{i=1}^{\infty} E_n$ then $\int_E f \le \sum_{i=1}^{\infty} \int_{E_n} f$'? You ...
4
votes
1answer
215 views

The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
15
votes
6answers
512 views

If $G$ is a group and $N$ is a nontrivial normal subgroup, can $G/N \cong G$? [duplicate]

I know $G/N$ is isomorphic to a proper subgroup of $G$ in this case, so the gut instinct I had was 'no'. But there are examples of groups that are isomorphic to proper subgroups, such as the integers ...
3
votes
2answers
184 views

finding a function with predetermined $f^{(n)}$s at $0$ and $1$.

Let $a_n,b_n$ be 2 arbitrary sequences of real numbers. Is there any $C^\infty$ function $f:\mathbb{R}\to\mathbb{R}$ such that for any $n$: $$f^{(n)}(0)=a_n,\quad f^{(n)}(1)=b_n$$
1
vote
2answers
243 views

Product of quotient maps and quotient space that is not Hausdorff

I am looking for easy(!) counterexamples that the product of two quotient maps is not necessarily a quotient map and that the quotient space of a Hausdorff space is not necessarily Hausdorff.
0
votes
1answer
20 views

Orthonormal basis for Hilbert space

Usually, an orthonormal basis for Hilbert space means an orthonormal subset which unconditionally spans the whole space. However, I'm curious whether there exists an orthonormal basis for every ...