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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Finding a counterexample: If $f=g^2$ and $f$ is differentiable on $[a,b]$, then $g$ is differentiable on $(a,b)$.

Suppose $f,g:[a,b]\to \mathbb R$. Provide a counterexample: If $f=g^2$ and $f$ is differentiable on $[a,b]$, then $g$ is differentiable on $(a,b)$. I've been attempting to find a counterexample by ...
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1answer
25 views

Borel Measures: Continuous vs. Discrete [on hold]

Here, it will be strictly distinguished between discrete and atomic!!! What is the rigorous definition for a Borel measure to be continuous? (The definition for discrete measure can be found on ...
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1answer
39 views

Counterexample of “the product of open subsets is open in a topological ring”?

Given a topological ring $R$ and $U,V$ open subsets, we can show that $U+V$ is an open subset due to the fact that $x\mapsto x+y$ is a homeomorphism for every $y \in R$. Since, in general, $R$ is not ...
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15 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) ...
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47 views

Serre fibrations vs. Hurewicz fibrations

What is an example of a Serre fibration that is not a Hurewicz fibration?
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2answers
91 views

Introductory example(s) of a functor that is full but not faithful

What is your favourite example to offer real beginners of a functor which is full but not faithful?
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1answer
27 views

On convergence a.e and convergence measure

I have a question. First, I know that convergence in measure of a sequence of functions $f_n$ is different than convergence a.e., wich means there are sequences that converge in measure but not a.e. ...
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1answer
44 views

A function $f$ such that the limit of $f(x^2)$ exists but not $f(x)$.

I want to show a function $f$ such that $\displaystyle\lim_{x\to x_0}f(x^2)\in\mathbb{R}$ but $\displaystyle\lim_{x\to x_0}f(x)$ doesn't exist. I only need a suggest of such a function $f$. I can't ...
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11 views

Quadratic stability linear time varying system

Consider the linear time-varying system $$ \dot{x} = A(t) x, $$ where $x \in \mathbb{R}^n$ and $A: [0,+\infty) \rightarrow \mathbb{R}^{n\times n}$ is continuous. It is known (see for instance, [1, ...
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20 views

Clarification on the definition of truth

So I am learning about the Godel's theorem. My instructor define truth and falsity as something arbitrary. he define $f$ to be true is $f(x)$ is $1$ and if $f(x) = 0$, it is false. I still dont have a ...
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39 views

Polynomial decomposition

I've just recently learned about the neat algorithm that, given a polynomial $f$ finds (non linear) polynomials $h,g$ such that $$f = g \circ h \quad (1),$$ or decides that there are no such ...
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3answers
47 views

Borel Sets which are not intervals

I am looking for an element of the Borel-sigma-algebra which is not an (open, closed, half-open,...) interval. Can someone provide any example or an algorithm to construct them?
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2answers
14 views

What would be an example that rank of a subgroup of a free group is greater than the rank of free group?

Let $G$ be a free group and $H$ be a subgroup of $G$. Then, $H$ is also a free group. Let $R_G,R_H$ be ranks of $G,H$ respectively. what would be an example that $R_H>R_G$?
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1answer
29 views

Constructing noncommutative nilpotent rings of given index

When I read about algebra I often see a certain disregard for examples or perhaps a disregard for a reader whose knowledge of examples is limited. When I'm interested in a property $p$ of an algebraic ...
1
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1answer
33 views

Atlases on the topological manifold $\mathbb R$

I have been trying to produce an example of two incompatible atlases on $\mathbb R$. But no success. Could someone help me please? All my example seem compatible. For example, $A_1 = \{((-\infty,1), ...
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4answers
75 views

What are non-obvious examples of measures obtained from linear functionals by the Riesz representation theorem?

In chapter two of Rudin's "Real and Complex Analysis" there is a "Riesz Representation Theorem" that dominates the chapter. My understanding of the statement of the thm. is that given a complex-valued ...
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3answers
49 views

Does a symmetric matrix $A^2$ imply a symmetric $A$?

Does a symmetric matrix $A^2$ imply a symmetric $A$? Any help would be much appreciated.
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1answer
22 views

Product Hilbert Spaces that require completion?

My lecturer said that to make the Hilbert space $(H\oplus H',⟨\cdot,\cdot⟩ )$ we need to (1) make the cartesian product $H \oplus H' = H\times H'$, (2) give it an inner product, and - the confusing ...
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1answer
45 views

Seeking a possible counterexample in probability.

I am trying to find a counterexample or prove the following: $\dfrac{Var\left(X_{n}\right)}{\left[EX_{n}\right]^{2}}\rightarrow0 , then \dfrac{X_{n}}{EX_{n}}\rightarrow1$ in probability. Assuming ...
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2answers
52 views

Suppose f : A ---> B and g : B ---> A are functions for which g o f = 1A…

If I were to suppose that $f : A \to B$ and $g : B \to A$ are functions for which $g \circ f = 1_A$, is $f$ always surjective and is $g$ always injective? How would I either prove this or counter it? ...
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1answer
20 views

Continuity and Subspace Topology

I think the first one is false. If we let $(-1/2, 1/2) \subset \Bbb R$ and $(0,1/4) \subset \Bbb R$, then for $f(x) = x$ defined on $[0,1) \subset M = \Bbb R$, we have $f^{-1}(-1/2, 1/2) = ...
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1answer
15 views

A net in a product space and its cluster point

Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be two topological spaces. And let $Z=X\times Y$ be the product space equipped with the natural product topology $\mathcal{T}_Z$ on $Z$. Then, let ...
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1answer
23 views

Counterexample for existence of a minimiser in a variational problem

I'm trying to find an example of a minimisation problem of the form $$ \inf \{ J(u) := \int_{\Omega} f(x)|u(x)| + |\nabla u(x)|^2:\, u \in H^1, \, \int u = 1\}$$ with $\Omega$ an open and bounded ...
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1answer
29 views

Ring subset which absorbs but is not an additive subgroup

Are there any undegrad-level examples of ring subsets which possess absorbtion property (as in ideal definition) but are not ideals (i.e. are not additive subgroups)?
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58 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.
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1answer
26 views

Example of ring with two maximal ideals such that the char of the quotients is $0$, respectively $p$.

I am looking either for an example of a commutative ring with identity and two maximal ideals, such that the characteristic of one of the quotient rings is finite and the other characteristic is zero, ...
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1answer
31 views

Stuck trying to find unbounded $s_n$ with $\frac{1}{n}\sum_{k=1}^n s_k\rightarrow L$

I proved that if a sequence $(s_n)$ converges to a limit $s$ then so does its "average sequence," $(\sigma_n)$ with $\sigma_n=\frac{1}{n}\sum_{k=1}^n s_k$. I found a counterexample for the converse, ...
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0answers
118 views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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1answer
40 views

Example of a ring $R$ such that $R\otimes_R R\not\simeq R$

I want to show the following statement: If a ring $R$ is commutative and $I,J\triangleleft R$, then $$ R/I\otimes_R R/J\simeq R/(I+J). $$ I can easily show this, assuming that $1\in R$. What can be a ...
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2answers
41 views

Example of differentiable function with $f'(s_{n})=0$ but $f'(0)>0$

Ex: Give an example of a differential function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $0$ is a limit (accumulation) point of a sequence of critic points ($f'(x)=0$) and however $f'(0)>0$ ...
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1answer
50 views

Calc I problem involving continuity [closed]

If $g(x)=(h(x))^2$, then give an example of a function $h:\Re\rightarrow\Re$ with the property that $h$ is not continuous at $x=0$ but $g$ is.
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1answer
42 views

Finding a function which is onto, monotone and not one-one [closed]

Does there exist a function $f:[0,1] \to [0,1]$ which is onto, monotone and not one-one?
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1answer
78 views

Is every subgroup of a simple group is itself simple [closed]

Is the statement True of false? Every subgroup of a simple group is itself simple.
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1answer
86 views

How to construct a smooth curve whose range is dense in $\mathbb R^2$?

How to construct a smooth curve $f: \mathbb R \to \mathbb R^2$ whose range is dense in $\mathbb R^2$? Space-filling curves are well-known, but they cannot be smooth. The image of a smooth ...
3
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1answer
60 views

A derivative which is not Lebesgue integrable on any interval?

If $f=x^2\sin(x^{-2})$, then $f'$ exists everywhere (including $x=0$) but $f'$ is not Lebesgue integrable on $[0,1]$ (precisely because of the singularity at $x=0).$ I'm trying to find a function $f$ ...
3
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1answer
42 views

Module over an associative ring with unity and axioms of projective geometry

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
2
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1answer
40 views

Proving the derived set $E'$ is closed.

I was reading the proof in Rudin, but it uses the metric. Is this not true if $X$ is a general topological space and $E' \subset X$ (especially if it is not Hausdorff $T_1$)? I can't come up with a ...
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0answers
32 views

Bochner vs. Lebesgue

I'm trying to prove that for complex functions $f:\Omega\to\mathbb{C}$ that are not a priori measurable that: $$f\text{ Bochner integrable}\iff f\text{ Lebesgue integrable}$$ Basically it reduces to ...
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1answer
34 views
0
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1answer
51 views

Example of a locally compact metric space whose completion is not locally compact

Can someone suggest an example of a locally compact metric space whose completion is not locally compact?
4
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1answer
47 views

Are primitive row stochastic matrices diagonalizable?

Let $A$ be an $n \times n$ matrix with real, non-negative entries. Assume $A$ is primitive, meaning there exists an integer $k$ such that $A^k>0$ (here the inequality means all entries in $A$ are ...
2
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2answers
41 views

Counterexamples for $f(\overline{A}) = \overline{f(A)}$ and $\overline{f^{-1}(B)} = f^{-1}(\overline{B})$ in (non-)continuous mapping $f: X \to Y$

Let $f$ be a mapping. Prove that the following three statements are equivalent. $f$ is continuous; $\forall A \subseteq X: f(\overline{A}) \subset \overline{f(A)}$; $\forall B \subseteq ...
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0answers
26 views

Basic calculus question with continuous function [duplicate]

This is actually not my question, it was asked yesterday by user176744 in this link $[0,n]$ continuous function problem and I feel as if it didn't get enough attention. I am also interested in a ...
1
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1answer
36 views

A Real valued function which is discontinuous **only** on a given specific set.

Let $\mathbb{L}=\{x_n \ |\ n=1,2,3 \dots\}$ be a countable subset of $\mathbb{R}$. My aim is to construct a real valued function $f$ on $\mathbb{R}$ such that $f$ is discontinuous at every point ...
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2answers
55 views

Vector space without a scalar product

In linear algebra the terms vector space and scalar product always (at least for me) appear together. Can you give me an example of a vector space without a scalar product? Does the senescence Let V ...
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2answers
63 views

Can the derivative of an absolutely continuous real function have a simple discontinuity?

If $f'$ exists everywhere, then we know that it cannot have any simple discontinuities. But in this case we only know that $f'$ exists a.e. (since $f$ is absolutely continuous). More specifically, ...
2
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1answer
48 views

Sets and Logic .. Disproving with counter example

Can anyone please give me an idea to disprove the following with a counterexample: $A , B , C$ are sets. If $A \times C = B \times C$ , then $A = B$. (Here $\times$ is a Cartesian product.) I tried ...
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2answers
60 views

Is it true that if $x_n$ converges and $y_n$ is bounded, then $x_ny_n$ converges?

Is it true that if $x_n$ converges and $y_n$ is bounded, then $x_ny_n$ converges? $x_n$ is said to be bounded if and only if it is bounded both above and below. I believe this to be false. My ...
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1answer
24 views

A function with only a partial derivative not Hölder-continuous

I'm looking for a function of two variables, say $u(t,x)$, such that for some $\alpha\in ]0,1]$ 1. $x\mapsto u(t,x)$ is $C^{2,\alpha}$; 2. $t\mapsto u(t,x)$ is $C^{1,\alpha}$; 3. $t\mapsto ...
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0answers
19 views

Examples of “finite character”

The property of being a linear ordering has finite character, i.e. a relation linearly orders a set if it linearly orders all of its finite subsets. This is a trivial corollary of the definition. ...