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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Interesting Problems for NonMath Majors

Sometime in the upcoming future, I will be doing a presentation as a college alumni to a bunch of undergrads from an organization I was in college. I did a dual major in mathematics and computer ...
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1answer
28 views

what would be an example of a function such that $\int_{|z|=r} f(z)=0$ for all $r>0$, but not analytic everywhere?

what would be an example of a function such that $\int_{|z|=r} f(z)=0$ for all $r>0$, but not analytic everywhere. I cannot think of one..
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1answer
19 views

Are surjective order-homomorphisms necessarily complete lattice homomorphisms?

Let $X$ and $Y$ denote complete lattice, and suppose $f : X \rightarrow Y$ is a surjective order-homomorphism. Does $f$ necessarily preserve arbitrary suprema, therefore being a complete lattice ...
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2answers
36 views

Does $(x-a)^n\in K[x]$ imply that $a\in K$?

Let $F$ be an extension field of $K$. Let $a\in F$ and $n$ be a positive integer. It is also given that the polynomial $(x-a)^n$ has all of its coefficients in $K$, i.e. $(x-a)^n\in K[x]$. Does it ...
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2answers
56 views

What are counter examples for these statements?

Question 1. Let $\{T_i\}_{i\in I}$ be a family topologies on a set $X$. Provide an example that $\bigcup T_i$ is not a topology on X. > Question 2. Let $X$ be a compact space ...
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0answers
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Can an algebraic structure such that $x+a=x+b$, have solutions for all $a,b∈K$ exist?

Is there exist a algebraic structure $K$ such that equations of the form $x+a=x+b$, $a\neq b$ have solutions for all $a,b∈K$?
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0answers
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Homeomorphisms of product spaces: an example [duplicate]

In the first of these lectures (http://www.mpim-bonn.mpg.de/node/4436) given by M. Freedman he says that there exists (compact metric) spaces $X$ and $Y$ such that $X\times S^{1}$ is homeomorphic to ...
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3answers
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Differential Equations without Analytical Solutions

In many talks, I have heard people say that the differential equation they are interested in has no analytical solution. Do they really mean that? That is: Can you prove a differential equation ...
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1answer
48 views

What is wrong with this “counterexample” of boundedness of weakly convergent sequences?

Weakly convergence sequences $\{u_n\}$ in a Hilbert space $H$ are bounded. Here is an attempted "counterexample". What is wrong with this? Let $H = \ell_2(\mathbb{N})$, and let $\{e_n\}$ be the ...
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1answer
19 views

Necessity of a hypothesis in Scheffé's lemma

Scheffé's lemma states that if $f_n$ is a sequence of Lebesgue integrable functions (i.e. $f_n \in L_1$) that converges almost everywhere to another integrable function $f$, then $\int |f_n - f| \, ...
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1answer
39 views

Product of two symmetric banded matrices - real eigenvalues?

Let $A$ and $B$ be real symmetric banded matrices but $AB$ is not symmetric. Are the eigenvalues of $AB$ real? A more specific case: let $D$ be a real diagonal matrix, $B$ real symmetric and banded, ...
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A locally connected metric space of first category

I wonder if there exists a locally connected metric space of first category. I proved a theorem which assumes a space to be connected locally connected hereditarily Baire and metrizable. I ...
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0answers
30 views

A real-valued separately continuous function discontinuous everywhere

I need an example of a real-valued separately continuous function that is discontinuous at each point. This is either a well-known folklore fact or a burning problem in separate versus joint ...
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0answers
17 views

F sigma subsets of the real line

I need an example of a nowhere dense $\mathcal F_\sigma$ subset of the real line that is not a countable union of perfects sets and not a countable union of pairwise disjoint closed sets.
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5answers
725 views

Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere?

Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere? I think it is probable because we can consider $$ y ...
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2answers
70 views

A function such that $f'(x)>0$, but not strictly monotone increasing.

This is Exercise 10.3.5 from Analysis Vol.1 by Terence Tao. Give an example of a subset $X \subset \mathbb{R}$ and a function $f: X \to \mathbb{R}$ which is differentiable on $X$, is such that ...
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1answer
47 views

Continuity conditions for multivariate functions.

Is the following true ? A proof or counter-example or reference would be nice. A function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ is continous at $(0,0)$ if and only if if for all $a, b$, the limits ...
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1answer
277 views

Example of similar matrices $A$ and $B$ such that products $AB$ and $BA$ are not similar

I'm looking for the simplest possible example of square matrices $A$ and $B$ such that $A$ is similar to $B$, $AB$ is not similar to $BA$. Such an example should exist, but I would like to find ...
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2answers
44 views

Topological vector spaces vs. locally convex spaces

I'm taking a course on locally convex spaces and our lecturer mentioned that these form the most general collection of spaces on which one can still prove interesting theorems (like Hahn-Banach - ...
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1answer
27 views

Extending a compact operator to the entire Hilbert space

In a course I'm taking we defined compact operators as a linear mapping $H\rightarrow H$, where $H$ is a Hilbert space, that maps bounded sets to relative compact ones. The lecturer mentioned that the ...
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1answer
113 views

Example where Čech and derived functor cohomologies don't agree.

I'm studying sheaf cohomology, and I've seen that Čech and derived functor cohomologies agree, at least on paracompact Hausdorff topological spaces. Is there a simple example of a topological space ...
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0answers
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is this a valid counter-example - function is not locally invertible

Let $S_n$ be the set of all symmetric matrices with real entries of size $n$x$n$. We are asked if the function $f:S_n \to S_n$, $f(A)=A^2$ is locally invertible for every $A$ (Using the Inverse ...
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2answers
83 views

Closure of Integers under multiplication and rational exponentiation

What is the closure of the Integers under a finite number of multiplications and rational exponentiations? For example, $3^{1/2}$, $i = -1^{1/2}$, and $\frac{-1+i \sqrt(3)}{2} = 1^{1/3}$ all in this ...
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0answers
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Counterexample for converse about measurable sections

On page 67 of Jacod and Protter, Probability Essentials, it is stated that: Theorem 10.2 Let $f$ be measurable: $(E \times F, {\cal E} \otimes {\cal F}) \to (\mathbf R, {\cal R})$. For each $x \in ...
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1answer
41 views

$\mathcal{C}^\infty$ strictly monotone function $\lim f(x) = 0$, but $\lim f^\prime(x) \ne 0$

Does there exist a strictly monotone function $f\colon \Bbb R\to\Bbb R$ that is $\mathcal{C}^\infty$, and $\lim\limits_{x\to+\infty} f(x) = 0$, but $$\lim_{x\to+\infty} f^\prime(x) $$ and ...
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2answers
27 views

Example of a subgroup of index two which contains a non square element

If a finite group contains a subgroup H of index two, then every element of the group which is a square belongs to H. Is there a (simple) counterexample showing that not all the elements of H are ...
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1answer
46 views

Counterexample to Marcinkiewicz

I have a version of Marcinkiewicz: Let $(X,\mu)$ and $(Y,\nu)$ be measure spaces and let $1<p_1 \leq \infty$. Suppose that $T$ is a mapping from $L^1(X,\mu) + L^{p_1}(X,\mu)$ to $\mu$- measurable ...
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1answer
326 views

Counterexample for the Open Mapping Theorem

I would like to ask a counterexample for the open mapping theorem: Find a discontinuous linear mapping $T:X \to Y$ such that $T(X)=Y$ and $X,\;Y$ are Banach but $T$ is not open. Could you help me ...
3
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1answer
97 views

Infinite abelian group counterexample [duplicate]

A finite group $G$ is abelian iff all its irreducible representation $\rho$ have dimension 1. I'm looking for a counter-example when $G$ is an infinite group. Are there any? EDIT We're dealing ...
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2answers
60 views

Question about a counterexample concerning compact operators

Does anybody know if the following is true, Let $H$ be an infinite dimensional Hilbert-space and $K:H\rightarrow H$ a compact operator. Then if $|\mathrm{spec}(K)|<\infty$ i.e the spectrum is ...
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1answer
72 views

counterexample for ill posedness of the laplace equation

Consider the wave equation with initial data: $$u_{tt}(t,x) + u_{xx}(t,x) = 0$$ $$u(0,x) = u_0(x)$$ $$u_t(0,x) = u_1(x)$$ Hadamard showed that this problem is ill-posed: there exist large solutions ...
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1answer
263 views

Open and Closed Quotient Maps.

a) Find a subset $A$ of $\Bbb{R}$ such that the quotient map $p: \Bbb{R} \rightarrow \Bbb{R}/A$ is not open. If we let $A= \Bbb{Q}$, then we can see that $(0,1)$ is open in $\Bbb{R}$. But if we ...
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1answer
34 views

Question about compact operator

So here is my question, Let $H$ be a Hilbert-space and $K:H\rightarrow H$ a compact operator. I know that if $K$ is self adjoint, then it has one eigenvalue $\mu$ such that $|\mu|=||K||$. Can some ...
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3answers
57 views

applications of the identity $ab + \left(\frac{a+b}{2} - b\right)^2 = \left(\frac{a+b}{2}\right)^2$

I am reading euclid's elements I found the algebraic identity $ab + \left(\frac{a+b}{2} - b\right)^2 = \left(\frac{a+b}{2}\right)^2$ I ponder on usage of this identity for $2$ hours. but I can't ...
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1answer
35 views

Extraneous solutions.

I just learned of extraneous solutions on the internet and thought, "could you both lose and gain solutions in one equation?" I think that, yes, you should be able to do that. However I haven't been ...
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2answers
43 views

Problem on Convergence of random series

Suppose that $\{X_n\}$ is an independent sequence and $E[X_n]=0$. If $\sum \operatorname{Var}[X_n] < \infty$, then $\sum X_n$ converges with probability $1$. Is independence necessary condition ...
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1answer
16 views

Counterexamples for the image of central,idempotent,invertible and nilpotent elements of a ring

I already proved that if i was given a surjective ring morphism f from R to S and then if a∈R is invertible, central, idempotent, or nilpotent, respectively then f(a) also is. But im looking for ...
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1answer
23 views

Do all distributions of R.V.s have a singular part and a continuous part?

Consider the probability distribution of a real-valued R.V. as the equivalence class of generalized PDFs where the integral over each measurable set in $\mathbb{R}$ is the same in each PDF. 1) Can ...
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1answer
27 views

Counterexamples for L1-convergence not implying L2-convergence and vice versa

I am trying to find counterexamples for the following statements. Let $\{f_n\}$ be a sequence in $L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$, and let $f$ also be in $L^1(\mathbb{R}^d) \cap ...
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8answers
2k views

Statements with rare counter-examples [duplicate]

This is a soft question. I'm searching for examples of mathmatical statements (preferably in number theory, but other topics are also fine), that seem to be true, but are actually not. Statements ...
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1answer
86 views

If partial sums of derivatives diverge then derivative does not exist?

Let $f_n :[a,b]\to \mathbb R$ be differentiable and such that $\sum_n f_n'(x)$ diverges for all $x \in [a,b]$. Let $\sum_n f_n $ converge uniformly on $[a,b]$ to $f$. Does it follow that $f'$ does ...
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0answers
29 views

an example that property $\delta$ does not imply property $\gamma$

In this article, two properties are mentioned at page 153: property $\gamma$: If $\mathcal U$ is an open $\omega$-cover of $X$, then there is a sequence $G_n \in \mathcal U$, with $\underline {Lim} ...
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1answer
50 views

Maps from subsets of $\mathbb{R}^2$ that are either open/closed/continuous

I'm self studying J. Lee's Introduction to Topological Manifolds, and after doing all other exercises on chapter 2, I can't seem to come with the proper counterexamples for this one. For each of ...
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1answer
44 views

Some examples of local and nonlocal properties

Today I learned that continuity at a point is a local property. Concretely, if $f: \mathbb R \to \mathbb R$ is continuous on $[-K,K]$ for all $K \in \mathbb R$ then $f$ is continuous on $ \mathbb ...
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1answer
20 views

Blowup of ODEs in the presence of local Lipschitzianity?

Pardon me if the question is trivial, but I am failing to decide it. Assume that we are given an ODE system $\dot{x} = f(x)$ with positive initial conditions $x(0)$ and know that $f$ is locally ...
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1answer
170 views

A counterexample

Let $f:\mathbb{R}\to(0,\infty)$ a locally integrable function. I want to compare these two conditions $$\limsup_{r\to + \infty}\frac{r}{\int_{-r}^r f(x)dx}<+\infty. \tag{1}\label{1}$$ and ...
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0answers
30 views

The closed graph theorem for Banach spaces isn't true. True?

I'm reading through some functional analysis lecture notes and there the closed graph theorem was stated in the following form: Let $X$ be a Baire locally convex space and $Y$ a Frechet space. If ...
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2answers
50 views

Example for a corollary in finite field theory

I'm preparing a seminar and the problem is that I never had a lecture before in finite fields. So I had to learn everything by myself, and that is unfortunately not easy at all... Can anyone help me ...
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1answer
41 views

weak star convergence of signed measures vs convergence in Fortet-Mourier norm

There is a norm for signed measures given by $$||\mu||_{FM}=\sup_{f\in \mathrm{Lip}_1(X),|f(x)|\leq 1}\langle f,\mu\rangle.$$ This is usually called Fortet-Mourier norm (or more often metric, but it ...
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1answer
43 views

L'Hospital's rule vs Taylor series

One classical application of Taylor expansions is to obtain polynomial equivalents of complicated functions and use them to compute limits. For example, with Landau notations, we have ...