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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2
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3answers
212 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 ...
7
votes
1answer
362 views

Example of a set $Y$ that has zero Lebesgue measure and a continuous function $f$ such that $f(Y)$ is not a set of zero Lebesgue measure.

Could someone give me an example of a set $Y\subset \mathbb{R}$ that has zero Lebesgue measure and a continuous function $f:X\subset \mathbb{R}\to\mathbb{R}$ such that $Y\subset X$ and $f(Y)$ is not a ...
4
votes
1answer
269 views

Example of a function that has the Luzin $n$-property and is not absolutely continuous.

The Banach–Zaretsky theorem (page 196) says that a continuous function $f:[a,b]\to\mathbb{R}$ of bounded variation is absolutely continuous if and only if $$E\subset I \text{ has zero Lebesgue ...
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1answer
52 views

Looking for a counterexample in convergence of random variables

I was wondering is it possible that a sequence of $k$-dimensional random variables, $\{\mathbf{X_n}\}$ converges componentwise, but not jointly? What I mean is ...
1
vote
1answer
22 views

Example of two closed disjoint set $X, Y$ so that $d(X, Y) = 0$

I am looking for an example of a metric space $M$ and non empty disjoint closed subsets $X$ and $Y$ such I that $d(X,Y)=0$, where $$d(X,Y)=\inf_{x\in X, y\in Y} d(x, y).$$ I’m thinking it might have ...
0
votes
1answer
338 views

Counterexamples for Borel-Cantelli

Our teacher mentioned to construct two counterexmaples for Borel-Cantelli using the following ways. (a) Construct an exmaple with $\sum_{i=1}^{\infty}\mathbb P(A_i)=\infty$ where $\mathbb ...
4
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1answer
57 views

Applications of the completeness of $L^1$

I'm teaching a measure theory class. I think one of the main motivations for the development of the Lebesgue integral is that the space $L^1(\mathbb{R})$ of integrable functions on $\mathbb{R}$ is ...
5
votes
1answer
591 views

Convergence types in probability theory : Counterexamples

I know that the following implications are true: $$\text{Almost sure convergence} \Rightarrow \text{ Convergence in probability } \Leftarrow \text{ Convergence in }L^p $$ $$\Downarrow$$ ...
2
votes
1answer
98 views

Examples of extremally disconnected spaces

I am trying to understand the notion of extremally disconnected space (in other words Stonean space), i.e. a space in which any open set has an open closure. Could you help me and give (reasonable) ...
1
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2answers
34 views

Family of subsets which is closed under finite disjoint unions & complementation, but not a field

I want to find an example of a family of subsets $\mathcal{F}$ of a set $X$ such that $X$ belongs to the family, it is closed under complementation and finite disjoint unions, but $\mathcal{F}$ is not ...
2
votes
2answers
726 views

Convergence of Monotone sequences? example

An example of an unbounded increasing sequence that satisfies the assumptions of the convergence of monotone sequences...? According to the convergence of monotone sequences if a sequences is ...
2
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0answers
23 views

function $f\notin L^{\infty}(\Omega)$, but $f\in L^p(\Omega)$ for all $1\le p <\infty$ [duplicate]

I'm searching for a function $f\notin L^{\infty}(\Omega)$, but $f\in L^p(\Omega)$ for all $1\le p <\infty$. And it has to be $|\Omega |<\infty$. I tried $f(x)=\frac{1}{2\sqrt{x}}$ and $\Omega= ...
10
votes
2answers
106 views

If $f:[a,b]\to \mathbb{R}$ satisfies $|f'(x)|<1, \forall x\in [a,b]$, is $f$ necessarily a contraction?

If $f:[a,b]\to \mathbb{R}$, $f'(x)$ exists for all $x\in [a,b]$ (derivatives at endpoints $a,b$ are one-sided) and satisfies $|f'(x)|<1, \forall x\in [a,b]$, is $f$ necessarily a contraction (i.e. ...
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9answers
4k views

Are there any situations in which L'Hopital's Rule WILL NOT work?

Today was my first day learning L'Hopital's Rule, and I was wondering if there are any situations in which you cannot use this rule, with the exception of when a limit is determinable.
2
votes
2answers
136 views

The preimage of a Lebesgue measurable set under a measurable function need not be measurable

I am reading measure theory from Royden, and I am stuck in some of them. I have this question: suppose $E$ is a measurable set and let $f: E \to \mathbb{R}$. Prove that : $f$ is measurable if ...
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0answers
21 views

Which functions are in $C_0^\infty$, except for the Bump function

We all know the standard example for a function in $C_0^\infty(\Omega)$, namely \begin{align} \Phi(x) = \begin{cases} e^{ -\frac{1}{1 - x^2}} & \mbox{ for } |x| < 1\\ 0 & \mbox{ otherwise.} ...
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votes
3answers
1k views

Continuity of the inverse $f^{-1}$ at $f(x)$ when $f$ is bijective and continuous at $x$.

Prove or disprove: Let $f:\mathbb{R}\to\mathbb{R}$ be bijective and $f$ is continuous at $x$. Then $f^{-1}$ is continuous at $f(x)$. Any hints are welcome. If this is false, I would like to have ...
2
votes
2answers
62 views

A surface with Euler characteristic of $-1$ [closed]

Is it possible to a have a surface that has an Euler Characteristic of $-1$ and what would that surface be homeomorphic to? $\displaystyle \chi \left({M}\right) = -1$
1
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1answer
84 views

2-dimensional Cohen-Macaulay domain

I am searching for a $2$-dimensional Cohen-Macaulay (normal or not) domain. Thanks in advance for any suggestion.
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5answers
1k views

Why does the Hilbert curve fill the whole square?

I have never seen a formal definition of the Hilbert curve, much less a careful analysis of why it fills the whole square. The Wikipedia and Mathworld articles are typically handwavy. I suppose the ...
11
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3answers
2k views

Is there an example for an undefinable number?

This question is motivated by a comment of Robert on the question Can any Real number be typed in a computer? : Can you "think of" an undefinable number? – Robert Israel I would like to ...
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1answer
65 views

Why can't Fubini's/Tonelli's theorem for non-negative functions extend to general functions?

Challenging-conventional-wisdom question based on an answer to my previous question. If $X \in L^1 (\Omega, \mathscr{F}, \mathbb{P})$ has pdf $f_X$, $Y \in L^1 (\Omega, \mathscr{F}, \mathbb{P})$ has ...
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2answers
4k views

Does giving a counterexample to a conjecture prove it to be true or false?

If a problem asks me to give a counterexample to a conjecture, am I proving the conjecture true or false by giving a counterexample? I am leening towards proving it false, because if I were to prove ...
2
votes
1answer
76 views

Is the conjugacy map between two distinct circle homeomorphisms unique?

Suppose $f,g,h_1,h_2$ are circle homeomorphisms with $f≠g$ and $fh_i = h_ig$ for $i=1,2$. Does it follow that $h_1 = h_2$? I restrict $f≠ g$ because I noticed that if $$f(x) := g(x) := R_\alpha(x) ...
1
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0answers
30 views

Analytic automorphisms of $n-$disk that fix the origin

Let's write an analytic surjection $f\colon D^n\longrightarrow D^n$ that fixes the origin $f(0)=0$, where $D^n=\{x\in\mathbb{R}^n\colon|x|\leq1\}$. Is it possible that $f$ is not an open map? ...
2
votes
1answer
50 views

Are partial derivatives always commutative? When is $\frac{\partial^2}{ \partial x\partial y}f(x,y)\neq\frac{\partial^2}{\partial y\partial x}f(x,y)$?

I learned in my Calculus 3 class that $\frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}f(x,y)\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial x}f(x,y)\right)$ Are there ...
3
votes
1answer
46 views

$E(X_n)=E(X_{n-1})=\ldots=E(X_0)\nRightarrow (X_n)_n$ is Martingale

I am searching for an example of an adapted process $(X_n)_n$ with constant expected value which is not an martingale (I know that the reverse direction holds)
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votes
2answers
1k views

Simply connected does not imply contractible. Is there a nice counter example in $R^2$?

The standard counter example to the claim that a simply connected space might be contractible is a sphere $S^n$, with $n > 1$, which is simply connected but not contractible. Suppose that I were ...
1
vote
1answer
34 views

counter example - affine space

In the affine-n-space $\mathbb A^n_k$ (where $k$ is algebraic closed) you can define for an algebraic set $X$: $I(X)=\left \{ f\in k[x_1,x_2,...,x_n] | \forall a \in X \,\,\,f(a)=0))\right \}$ I ...
2
votes
1answer
45 views

Isomorphism $\phi:G\to G\times G$ [duplicate]

Can anyone provide me with an example of a non-trivial group $G$ which is isomorphic to $G\times G$. What is the mapping $G \to G\times G$?
4
votes
1answer
117 views

Examples of bi-implications ($\Leftrightarrow$) where the $\Rightarrow$ direction is used in the proof of the $\Leftarrow$ direction.

[I'm asking for examples of proofs with a certain structure. There is quite a lot of text before arriving at the questions. This is because asking for examples of a phenomenon is best carried out by ...
1
vote
3answers
130 views

Example of an unbounded sequence whose convergent subsequences converge to same limit

"A bounded sequence of real numbers converges to x if every convergent subsequence of the sequence converges to x." I require a counterexample to prove that the theorem fails if the hypothesis that ...
50
votes
10answers
2k views

Fake induction proofs

Question: Can you provide an example of a claim where the base case holds but there is a subtle flaw in the inductive step that leads to a fake proof of a clearly erroneous result? [Note: Please do ...
0
votes
2answers
62 views

Finding $a_n\to 0$ such that $\sum a_n/n$ diverges [closed]

Can someone please help me find a counter example for the following claim? If $a_n \to 0$ then $ \sum_ 1 ^{\infty} \frac{a_n}{n}$ converges.
1
vote
0answers
50 views

Counter-examples in representations of associative algebra

Here are something well-known. $V \cong W \Rightarrow \chi_V=\chi_W$ holds for finite representations of arbitrary associative algebra. But $ \chi_V=\chi_W \Rightarrow V \cong W $ is true only for ...
2
votes
2answers
101 views

An example of finite, connected topological group

A finite Hausdorff topological group, has discrete topology and every discrete group is totally disconnected. I look for an example of a non-Abelian, finite, connected non-Hausdorff group . I think ...
194
votes
32answers
10k views

Can't argue with success? Looking for “bad math” that “gets away with it”

I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare"). One example would be "cancelling" the 6s in $$\frac{64}{16}.$$ ...
12
votes
5answers
1k views

Are continuous functions monotonic for very small ranges?

So I am wondering, if we have a continuous function $f : A \to B$, does a range $[x, x + h]$ exist for each $x\in A$ , $h = h(x)>0$ so that $f$ is monotonic in that range?
24
votes
3answers
536 views

Connected, locally connected, path-connected but not locally path-connected subspace of the plane

I am looking for a set in the plane (with respect to the natural Euclidean topology) that is connected, locally connected, path-connected but not locally path-connected. I did not find one in ...
0
votes
1answer
44 views

Disconnectedness, completeness and compactness.

I am in search of examples of metric space which is 1) Complete but not compact 2) Not complete but disconnected 3) Connected but not Complete 4) Compact but not connected. 5) Complete but not ...
4
votes
2answers
471 views

Give an example of a non-separable subspace of a separable space

I'm trying to find an example of a non-separable subspace of a separable space. What kind of examples are there?
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2answers
23 views

Disproving Differentiable Functions with Counter Examples

How do I find the counter example for... If a function $g$ is differentiable at $a$ and a function $f$ is not differentiable at $g(a)$, then the function $f \circ g$ cannot be differentiable at $a$.
21
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7answers
3k views

Perfect set without rationals

Give an example of a perfect set in $\mathbb R^n$ that does not contain any of the rationals. (Or prove that it does not exist).
4
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2answers
1k views

$(0,1)\to\mathbb{R}^2$ injective, continuous, not a homeomorphism on the image

Consider the map $$\gamma\colon (0,1)\to\mathbb{R}^2,\ t\mapsto (\cos(2\pi t),\sin(2\pi t)).$$ This is an example of a map which is continuous and injective but not a homeomorphism onto the image, ...
5
votes
1answer
554 views

When the sum of independent Markov chains is a Markov chain?

I try to find as much as possible cases, when the chain $Z(t) = |X_1(t)-X_2(t)|$ is Markov, where $X_1(t)$ and $X_2(t)$ are independent, discrete-time and space, preferably non-homogeneous Markov ...
4
votes
4answers
424 views

Give an example of a nonabelian group in which a product of elements of finite order can have infinite order. [duplicate]

So, I let a,b be elements in such a group. So |a|=n and |b|=m, n and m are finite. But |ab| needs to be infinite, but since |ab|=lcm(n,m), how can that be possible?
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5answers
3k views

Examples and further results about the order of the product of two elements in a group

Let $G$ be a group and let $a,b$ be two elements of $G$. What can we say about the order of their product $ab$? Wikipedia says "not much": There is no general formula relating the order of a ...
5
votes
6answers
1k views

Example of a group where $o(a)$ and $o(b)$ are finite but $o(ab)$ is infinite [duplicate]

Let G be a group and $a,b \in G$. It is given that $o(a)$ and $o(b)$ are finite. Can you give an example of a group where $o(ab)$ is infinite?
0
votes
1answer
57 views

conjecture in probability, assume conjecture to be true

there is a list of disproved mathematical ideas https://en.wikipedia.org/wiki/List_of_disproved_mathematical_ideas does anyone know a conjecture in probability theory which was first thought to be ...