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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4
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2answers
144 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
76 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 $f$ ...
5
votes
5answers
526 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
59 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
189 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 ...
1
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2answers
131 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
939 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
28 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 ...
2
votes
1answer
49 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 ...
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
124 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 ...
0
votes
1answer
36 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 ...
6
votes
1answer
103 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
28 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
31 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 ...
2
votes
0answers
30 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
75 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
58 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
112 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
58 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
314 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
105 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
38 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 ...
1
vote
2answers
102 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
51 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
52 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
24 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
50 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 ...
0
votes
1answer
40 views

How do I prove the Poisson integral formula for a harmonic function on exterior of a disk?

Let $R>0$ and $u:\mathbb{C}\setminus B(0,R)\rightarrow \mathbb{R}$ be a continuous function such that $u$ is harmonic on $\mathbb{C}\setminus\overline{B(0,R)}$. Assume that $u$ is bounded at ...
0
votes
2answers
31 views

Question Regarding the Commutativity of F-Algebras when the Algebra is finite dimensional over F.

Let $A$ be some $F$-Algebra, for some field $F$, with the property that $A$ is finite dimensional over $F$. Is $A$ always commutative?
0
votes
1answer
43 views

Counter example for - product of general cardinal separable spaces

I am looking for a counter example for the claim that a product (of any cardinal) of separable spaces is separable, I saw in Uncountable product of separable spaces is separable? and On the ...
1
vote
1answer
51 views

Finding a continuous $f\in C(\mathbb{T})$ with $\{\hat{f}(n)\}\notin \ell^p$

Denote the circle by $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Find an example for a continuous function $f\in C(\mathbb{T})$ with coefficients of the fourier series, $\{\hat{f}(n)\}\notin\ell^p,\forall ...
0
votes
1answer
33 views

What is an example of such series?

Related:Why is this sequence uniformly convergent? Let $K$ be a compact subset of $\mathbb{C}$. Let $f_n$ be a sequence of continuous functions such that $f_n:K\rightarrow \mathbb{C}$. Assume that ...
2
votes
2answers
52 views

Normed algebra with bounded multiplication

Sometimes one finds the following definition of a normed algebra: This is an algebra with a norm on the underlying vector space such that there is a constant $K \geq 0$ such that $|x \cdot y| \leq K ...
1
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1answer
62 views

Category theory: do other examples of “resplendent” properties exist?

Call a predicate $P$ defined on categories resplendent iff it satisfies the following condition: for all categories $\mathbf{D}$, if $P(\mathbf{D}),$ then for all categories $\mathbf{C}$, we have ...
2
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0answers
42 views

What's the name of this extremely common but extremely pathological continuous function?

Okay, so let's define a random function $F$, such that the value of $F(x)$ is uniformly distributed on $[-1,1]$, and such that for any $x$ and $y$ with $x \ne y$, $F(x)$ and $F(y)$ are independent. ...
0
votes
1answer
19 views

What would be a non-injective holomorphic function on $B(0,\epsilon)$?

Let $\epsilon > 0$. What would be a non-injective holomorphic function on $B(0,\epsilon)$ such that $f'(0)\neq 0$? Since $f'(0)\neq 0$, there exists a neighborhood of $0$ such that $f$ is ...
2
votes
1answer
21 views

Determining if a set is measurable by upper and lower sets

I have the following question regarding Lebesgue measure: If $A,B$ are measurable sets and I have $m(A\setminus E)=0$ and $m(E\setminus B)=0$, is it enough to determine that $E$ is measurable? We do ...
0
votes
1answer
41 views

If $\int_a^x f(t)\, dt$ is differentiable, is its derivative integrable?

Let $f$ be a real-valued Lebesgue integrable function on $[a,b]$. If $F(x) = \int_a^x f(t)\, dt$ is differentiable on $[a,b]$, is $F'(x)$ (Lebesgue) integrable? I know there are examples of ...
0
votes
0answers
35 views

There is a function with this condition…

Show that there is a function $f : P(\Bbb{N} ) \to \Bbb{N}$ ,( $P(\Bbb{N})$ is power set) with this condition that for $A \in P(\Bbb{N})- \{\emptyset \}$ , $f(A) \in A$ , is there function with this ...
2
votes
2answers
96 views

If $|a^{2}|=|b^{2}|$ then $|a|=|b|?$

If $|a^{2}|=|b^{2}|$ (for non identity elements $a$ and $b$ of a group $G$ and $|a|$ denotes the order of the element $a$) prove or disprove that $|a|=|b|.$ I tried as follows Clearly infinite order ...
1
vote
2answers
36 views

Example of an infinite non-normal subgroup $H$ of non-abelian (topological)group $G$.

I am trying to find Example of an infinite non-normal subgroup $H$ of non-abelian (topological)group $G$. For instance, The subset $\mathrm{T}(n,\mathbb{R})\subset \mathrm{GL}(n, \mathbb{R})$ of ...
2
votes
1answer
34 views

Uniform convergence does not closed under multiplication

Construct sequences $\{f_n\}$, $\{g_n\}$ which converge uniformly in some set $E$, but such that $\{f_ng_n\}$ does not converge uniformly on $E$ (of course, $\{f_ng_n\}$ must converge on $E$). My ...
7
votes
2answers
448 views

Is a continuous function locally uniformly continuous?

Assume a function, $f : X \to Y$, mapping between two metric spaces, $X,Y$, is pointwise continuous, i.e. for every $\varepsilon >0$ and $x \in X$ there exists a $\delta>0$ such that $$ ...
2
votes
1answer
45 views

Example of quasi-compact, non-quasi seperated scheme where qcqs fails?

The qcqs lemma (in Ravi Vakil's notes) says that if $X$ is a quasi-compact (qc) and quasi-separated (qs) scheme, for any global section $f$, the natural map from $\Gamma(X, O_X)_f \to \Gamma(X_f, ...