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
1answer
28 views

Construct an example where x(t, x_0) is bounded but limt→+∞ x(t, x_0) does not exist.

Suppose we are given an IVP $x = f(x), x(0) = x_0 $, x ∈ R^n for which we know that the unique solution x(t, x_0) exists globally in time. Construct an example where x(t, x_0) is bounded but limt→+∞ ...
3
votes
1answer
72 views

Does there exist a Hausdorff group which is not locally compact?

A topological space is countably compact if every countable open cover has a finite subcover. A topological space $X$ is locally compact if any point has a neighbourhood which is compact. A ...
1
vote
1answer
62 views

Examples and counter-examples for rings

Here's what I am trying to do: Listing mnemonics used: $D_n(\Bbb R)$ to denote diagonal $n \times n$ matrices with real coefficients. I.D. - Integral domain, E.D. Euclidean Domain Much of the ...
3
votes
3answers
112 views

If G is open and dense subset in $\mathbb R$ then show that $G\setminus\{x\}$ is also open and dense in R. is it true in general metric space?

If $G$ is open and dense subset in $\mathbb R$ then show that $G\setminus\{x\}$ is also open and dense in $\mathbb R$. is it true in general metric space? I know as $G$ is open and singleton set $\{...
-1
votes
1answer
40 views

Example of two analytic functions that differ at countably infinity many point

$\displaystyle f_1(x) = \frac{x^n-1}{x-1}$ and $f_2(x) = x^{n-1} + \cdots + 1$ have the same values except at $x=1$ (where $f_1$ fails to be analytic ). Is there an example of two analytic function ...
6
votes
3answers
267 views

Example of an uncountable dense set with measure zero

As stated in the title, I am trying to find an example of an uncountable dense subset of $[0,1]$ that has measure zero. My intuition is that such a subset cannot exist, but I do not have a proof of ...
0
votes
0answers
24 views

$z_0$ is an essential singularity of $f$-> is $z_0$ is an essential singularity of $\frac{1}{f}$?

Let $f:B_{\epsilon}(z_0)\setminus \{z_0\}\to \mathbb{C}$ holomorphic, without zero points and $z_0$ is an essential singularity of $f$. Question: Does $\frac{1}{f}:B_{\epsilon}(z_0)\setminus \{z_0\}\...
0
votes
1answer
68 views

Spans of subsets and union of two sets

Is this true or not? How would I prove or disprove this? If the set of vectors $\{a_1 \dots a_n\}$ spans a subset $S$ and the set of vectors $\{b_1 \dots b_n\}$ spans a subset $T$, then $\text{Span}...
0
votes
0answers
111 views

I feel like this cannot be proven. Am I setting up the contrapositive correctly?

The question ask: Use proof by contrapositive to show that if a positive integer is the product of  two distinct primes, then its square root is irrational. So I have not(q) -> not(p) as follows: ...
4
votes
1answer
93 views

Example of a subnet that have no subsequence.

I have an elementary question on nets because I'm not familiar with this concept. Here are two basic facts: Every subsequence of a sequence is a subnet; Not every subnet of a sequence is a ...
5
votes
1answer
47 views

Recreational conjecture on factoring groups

Consider the following: For a group $G$ with identity $e$, define $s: G \to \mathbb{N} \cup \{ \infty \}$ by $s(g) = \min \{ k \in \mathbb{N} : g^{k} = e \}$, where $ \min \emptyset = \infty$. ...
2
votes
2answers
38 views

Measure on a countable set

Is there a decent characterization of measure on an infinite countable set? At page 7 of "Introduction to Measure Theory and Integration" (Ambrosio, Da Prato, Mennucci), example 1.10 I found that "...
12
votes
1answer
190 views

If $R[x]$ and $R[[x]]$ are isomorphic, then are they isomorphic to $R$ as well? [duplicate]

There are examples of commutative rings $R \neq 0$ such that $R[x]$ is isomorphic to $R[[x]]$ (see this question; an example would be $R=S[x_1, x_2, \ldots][[y_1, y_2, \ldots]]$, with $S \neq 0$ any ...
4
votes
1answer
125 views

Example of a Borel measure, which is not Borel-regular

I have asked a question to find four types of outer measures here, and I could find three of the four examples. We call an outer measure $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ Borel, if ...
0
votes
1answer
19 views

Metric space with two similar points which are not in the same orbit.

Is there an example of a metric space $X$ with two points $p$ and $q$ so that for every $r>0$ the ball with radius $r$ and center $p$ is isometric to the ball with radius $r$ and center $q$ and yet ...
3
votes
0answers
37 views

What could be examples at calculus or introductory analysis level for the idea contained in the statement by David Hilbert?

I read the following quote in the book "As opposed to abstraction the art of doing mathematics consists in finding special cases which contain all the germs of generality. --David Hilbert", however ...
1
vote
1answer
16 views

Valid method to obtain a basis of a topological subspace?

Let $(X,\tau)$ be a topological space and $Y \subset X.$ We know that if $\mathcal{B}$ is a basis for $\tau$ and $\tau_{\small{Y}}$ is the subspace topology on $Y$, then we can obtain a basis for $\...
4
votes
2answers
151 views

Prove or Disprove there is a sequence $f_n$ of continuous function on [0,1] such that for each x $ \in [0,1] $, $f_n(x)$ converges to $f(x)$

Prove or disprove: If $f$ is non-decreasing real valued function on $[0,1]$ then there is a sequence $f_n$ of continuous function on $[0,1]$ such that for each x $ \in [0,1] $, we have $f_n(x)$ ...
4
votes
1answer
40 views

Definite integrals that are hard using the FTC but doable from first principles

When I was introduced to integration, I was briefly exposed to the definition of a Riemann sum, and as an illustration we worked out a couple of definite integrals directly from this definition: $$ \...
-1
votes
1answer
83 views

Counterexamples on homotopy equivalence and infinite product [closed]

Let $(X,a),(Y,b)$ be pointed spaces and $f:(X,a)\rightarrow (Y,b)$ be a continuous function. If the natural homomorphism $f_*:\pi_1(X,a)\rightarrow \pi_1(Y,b)$ is a group isomorphism, is $f$ ...
6
votes
5answers
548 views

Example of a relation that is reflexive but not symmetric

By definition, $R$, a relation in a set X, is reflexive if and only if $\forall x\in X$, $x\,R\,x$, and $R$ is symmetric if and only if $x\,R\,y\implies y\,R\,x$. I think $x\,R\,x$ can also be ...
1
vote
1answer
66 views

Direct sum of two closed subspaces of Banach space is not closed

I'm looking for an example of two closed subspace of a Banach space (or even a Hilbert space) whose sum is not closed. We have $l^2$ as Banach space and $A$ and $B$ are closed subspaces of $l^2$ : $...
3
votes
1answer
30 views

Example of a function which demonstrates this equivalent condition for continuity

Please note that I am not asking for a proof, just a confirmation of my understanding or a counterexample to the question I posed. We know that a function $f:M\to N$ for $M, N$ metric spaces is ...
10
votes
2answers
200 views

Open set containing rationals but complement non-denumerable

I am taking Real Analysis classes and I got a homework that asks me: Give an example of an open set $\mathcal{A}$ such that $\mathcal{A}\supset\mathbb{Q}$ but $\mathbb{R}-\mathcal{A}$ is non-...
1
vote
2answers
138 views

Example that differentiable functions do not form an integral domain?

Let $G$ be an open connected subset of $\mathbb{C}$ and $f,g$ be holomorphic functions on $G$ such that $fg=0$. If neither $f$ nor $g$ is zero over $G$, since $f$ and $g$ have countably many zeros ...
27
votes
3answers
950 views

What is the largest set for which its set of self bijections is countable?

I recently came across a problem which required some knowledge about the self bijections of $\mathbb{N}$, and after looking up how to construct some different bijections I came across the result that ...
2
votes
2answers
30 views

Examples of product of two $L^1_\text{loc}$ functions that is not $L^1_{\text{loc}}$

Let $f\in L^1_\text{loc}$ and $g\in L^1_\text{loc}$, does $fg \in L^1_\text{loc}$? My textbook says it isn't in a general case. However if $g\in \mathcal{E} = \mathscr{C}^\infty$, then $fg\in L^1_\...
2
votes
1answer
52 views

Easy examples of the Arzela-Ascoli Theorem

Let $X$ be a compact metric space. $M \subseteq C(X)$ is relatively compact if and only if $M$ (i.e. its elements) is equicontinuous and uniformly bounded. I've been told that this theorem gives me ...
1
vote
1answer
24 views

Unbounded operators: product of adjoints strictly extended by the adjoint of product

It is well known that, if $T,S$ and $ST$ are densely defined operators on a Hilbert space $H$, then $T^* S^* \subset (ST)^*$. The proof of this is easy. Moreover, it's readily seen that equality ...
8
votes
1answer
127 views

Counterexample of polynomials in infinite dimensional Banach spaces

I'm trying to prove exercise I.3.B in Mujica's "Complex analysis in Banach spaces". DEFINITIONS: A map $P$ is an m-homogeneous polynomial from $E$ to $F$ if there is a m-linear map $A$ from $E^m$...
1
vote
1answer
38 views

When checking the absolute continuity of a measure, is it enough to consider a generating algebra?

Let $(X,M)$ be a measurable space, and $M=\langle A \rangle$ in which $A$ is an algebra on $X$. Suppose that $v$ is a signed measure and $m$ is a positive measure on $(X,M)$. Now, can we say: $v$ is ...
2
votes
3answers
97 views

Finding an example of a set $G$ which is not a group

Suppose $G$ is a set and $\cdot$ is a binary operation on $G$ such that there exists an $e\in G$ such that $a\cdot e=a$ for a in $G$ and given $a\in G$, there is a $y(a)\in G$ such that $y(a)\cdot a=e$...
6
votes
1answer
106 views

Finding a space with $X \cong X+2$ and $X \not\cong X+1$.

Question. Is there a topological space $X$ with $X \cong X+2$ and $X \not\cong X+1$? Here, $X+n$ denotes the disjoint union (i.e. coproduct) of $X$ with $n$ isolated points. This question is similar ...
0
votes
2answers
31 views

What is an example of a nonconstant subharmonic function that attains a minimum?

Let $D$ be a domain in $\mathbb{C}$. What would be an example of a nonconstant subharmonic function that attains its minimum in the domain?
1
vote
0answers
32 views

Positivity of this improper integral

I am not relieved of the following problem and trip. Let $\varepsilon>0$ be a small parameter, $a>0$ be a given constant, $x_{\varepsilon}\in(0,a)$ be a given sequence such that $x_{\varepsilon}...
2
votes
0answers
34 views

A question about a case where Central Limit Theorem doesn't apply

I'm trying to read Lehmann's "Elements of Large Sample Theory" and I have the following question about the text. The classical Central Limit Theorem is stated as: Now, the author goes on to ...
3
votes
3answers
84 views

Looking for pathologic counterexample: Nonzero harmonic function which is zero on the unit circle except 1

From my Complex Variables class: Let $C_1$ be the unit circle, $B_1$ the open unit disc and $\Gamma = C_1 \backslash \{1\} $. I am looking for a nonzero function $u \in C(B_1 \cup \Gamma)$ which is ...
3
votes
1answer
325 views

Is the property “Existence of Antiderivatives” preserved under multiplication and composition?

Since differentiation is linear, we therefore have that if $f, g: I\to \mathbb{R}$ have antiderivatives (where $I\subset \mathbb{R}$ is an interval), then so does their linear combination. What if we ...
8
votes
1answer
116 views

Is a path-connected bijection $f\colon \Bbb{R}^n \to \Bbb{R}^n$ continuous?

While thinking about this question I was asking myself if a path-connected bijection $f\colon \Bbb{R}^n \to \Bbb{R}^n$ has to be continuous for $n>1$? If we drop the requirement that $f$ is ...
0
votes
1answer
59 views

A bounded function having I.V.P. but not Riemann Integrable.

I am searching an example of a function $f$ on $[a,b]$ such that $f$ is a bounded function having intermediate value property but is not Riemann Integrable on $[a,b].$ Please give me such type ...
8
votes
1answer
319 views

Continuous functions and uncountable intersections with the x-axis

Let $f : \mathbb{R} \to \mathbb{R}$ such that the set $X = \{x \in \mathbb{R} : f(x) = 0\}$ does not contain any interval (i.e. there is no interval $I \subset X$) Of course the set $X$ can be ...
2
votes
0answers
109 views

Game theory, Book by Tirole and Fudenberg, Never a weak best response,unclear example

In this book, I have the following problem: on page 446, there is a sentence: Note that $(0.9,0.9)$ is not removed by NWBR, as D is not dominated after C is deleted. I do not understand this "as". ...
2
votes
1answer
42 views

Large deviation theory--examples of irregular sets

Let $(X,\tau)$ be a topological space, let $\mathcal{B}$ be its Borel $\sigma$-algebra, and $\mu_\epsilon$ be a family of probability measures on $(X,\mathcal{B})$. Suppose also that $\mu_\epsilon$ ...
1
vote
3answers
76 views

$f'$ changes strict monotonicity but $f''$ isn't of strictly opposite signs

Let $I \subset \mathbb{R}$ be an open interval and $f \in C^2(I,\mathbb{R})$. I am looking for an (simple) example of $f$ with the following properties ($x_0 \in I$) $f'$ is strictly monotonic ...
2
votes
2answers
106 views

Is there a well known nontrivial counterexample to this claim?

Suppose we have $A\subseteq\mathbb{N}$ with the property that if $B\subseteq\mathbb{N}$ and $B$ is finite, then $\exists a\in A\setminus\{ 1\}$, $\forall b\in B$, $\gcd(a,b)=1$. Are there any well ...
2
votes
1answer
63 views

Are there functions that are Holder continuous but whose variation is unbounded?

I have recently been introduced to the concept of Holder condition and I was told that there are functions that are Holder continuous but whose variation in unbounded. Can anyone present an example, ...
7
votes
2answers
53 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
1answer
17 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 ...
0
votes
1answer
26 views

Newton's Method in unconstrained optimization fails to converge

In order to show that Newton's method can produce a sequence of iterates that diverges, an example given in my book is apply Newton's Method to minimize $f(x)={2\over 3}|x|^{3\over 2}$. starting at ...
2
votes
2answers
58 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 ...