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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1answer
19 views

Integration obeys countable subadditivity?

Does Lebesgue integration have the property of countable subadditivity: 'if $f$ is integrable on $E$ and $E = \bigcup_{i=1}^{\infty} E_n$ then $\int_E f \le \sum_{i=1}^{\infty} \int_{E_n} f$'? You ...
0
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2answers
19 views

Example of a function $u \in W^{1,p}(\Omega)$ whose extension $\hat{u}$ to be $0$ outside $\Omega$ $\hat{u} \notin W^{1,p}(\mathbb{R}^n)$

Let's consider a function $u \in W^{1,p}(\Omega)$, where $W^{1,p}(\Omega)$ is the Sobolev Space and $\Omega$ is an open set. When we extend $u$ to $\hat{u}$ like this: ...
0
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1answer
17 views

Orthonormal basis for Hilbert space

Usually, an orthonormal basis for Hilbert space means an orthonormal subset which unconditionally spans the whole space. However, I'm curious whether there exists an orthonormal basis for every ...
6
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0answers
53 views

A List of Standard or “Cliche” Homeomorphisms [duplicate]

Learning topology has been hard. I just cannot see how some people can come up with complex functions that link one space to another, in a homeomorphic sense. The explanations are always "Well if you ...
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0answers
41 views

Hard counterexample to the fact that outer measure is additive

My question is very short: does there exist a couple of sets $A,B\subset[0;1]$ such that $A\cap B=\emptyset$, but $\mu(A)=\mu(B)=1$? Here $\mu(\cdot)$ is outer measure. It's easy to construct ...
1
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0answers
15 views

A illuminating theorem on general modules over a ring

I'm going to do a one hour presentation on modules, free modules and projective modules over a ring. While most of my motivation for studying them comes from my interest in starting homological ...
3
votes
1answer
26 views

A smooth function which is nowhere real analytic, and preserves rationality of its argument

There are examples $\!^{[1]}$$\!^{[2]}$ of continuous infinitely differentiable (class $C^\infty$) functions $\mathbb R\to\mathbb R$ that are nowhere real analytic. I wonder if it is possible to ...
16
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2answers
2k views

Does there exist a complex function which is differentiable at one point and nowhere else continuous?

Let $f\colon\mathbb{C}\to\mathbb{C}$. We know that if $f^{\prime}(a)$ exists for some $a\in\mathbb{C}$ then $f$ is continuous at $a$. This is because, from the definition of the derivative, ...
1
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1answer
37 views

two ways of counting

I'm reading Morris DeGroot's Probability and Statistics. In chapter 1.9 there's an example 1.9.3 says that suppose that 12 dice are to be rolled. We shall determine the probability $p$ that each ...
2
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2answers
36 views

Sequence of bounded Operators (Is this a counterexample?)

I've to proof the following statement Let $X,Y$ be to banach spaces and $(T_k)_{k \in \mathbb{N}} \subseteq L(X,Y)$ a bounded sequence of bounded linear operators. Further it exists a dense subset ...
0
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0answers
10 views

Proving not an equivalence relation -the basic case

For the basic case Let $X=Y= \mathbb{R}$ and $R(X,Y)= \{(x,y) \in X \times Y : y=x^{2} \}$. I know it's not symmetric, not reflexive, not transitive. How do I provide a counterexample that it's not ...
0
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1answer
25 views

Example of a field extension $K/F$ such that $K$ is the splitting field of some (non-separable) polynomial in $F$, but not Galois over $F$?

An extension $K/F$ is Galois if and only if $K$ is the splitting field of some separable polynomial over $F$. Is there an example of some non-Galois extension $K/F$ where a (necessarily) non-separable ...
1
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1answer
31 views

Results on “subtraction” of measures and outer measures?

Most results I have seen involves addition of measures For example, let $m^*$ and $m$ be Lebesgue outer measure and Lebesgue measure respectively, then given $A = \bigcup\limits_{n = 1}^\infty E_n, ...
4
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3answers
38 views

Proof intersection is finite and non-empty

Course: Analysis (1st year course). Question: If $A_3$ is a subset of $A_2$ and $A_2$ is a subset of $A_1$ and so on... are all finite, nonempty sets of real numbers, then the intersection ...
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0answers
18 views

Is the set of differentiable points of a monotonic function Borel-measurable?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a monotonically increasing function. Then, it has a countable discontinuity and is differentiable almost everywhere with respect to the Lebesgue measure. ...
0
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1answer
73 views

need an example for an ode system with 3 limit cycles

I was trying to find an ode system in predator-pray model with at least 2 limit cycles and different foci but I had trouble finding any, does anybody have an example in mind? thanks in advance
0
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2answers
39 views

Is there any function in $L^2$ that is not integrable? [duplicate]

I know that there are functions in $L^2$ that are integrable but not continuous. Is there any function in $L^2$ that is not even integrable?
0
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0answers
21 views

example of quasi-transitive relation that is not transitive

I'm trying to come up with an example of a relation that is quasi-transitive but not transitive. The relation $ x R y $ is a subset of the cartesian product $XxX$, and the asymmetric relation is $xPy ...
0
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0answers
18 views

Example of continuous, integrable function with non-integrable slices

What is an example of a continuous and integrable function $f:\mathcal{R}^2 \to \mathcal{R}$ with infinitely many $y\in \mathcal{R}$ such that $g^y:\mathcal{R} \to \mathcal{R}$ defined by $g^y(x) = ...
0
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1answer
31 views

A nonmeasurable function such that $|f|$ is measurable, and the preimage of every point is measurable [duplicate]

Give an example of nonmeasurable function $f:(\mathbb{R}, Leb)\rightarrow \mathbb{R}$ such that $|f|$ is measurable and for every $a\in \mathbb{R}$ , $f^{-1}(\{a\})$ is a measurable set My ...
0
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2answers
35 views

A non metric first countable topological space [duplicate]

Every metric space is first countable, but what about the converse? Does it always hold? If not, can anyone give a counterexample? Thanks
1
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1answer
25 views

Can we get negative variance when calculating it for a small dataset using a naive formula?

In Knuth's Volume 2 Seminumerical Algorithms, chapter 4.2.2 Accuracy of Floating Point Arithmetic, there's a statement: Novice programmers who calculate the standard deviation of some observations ...
3
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1answer
60 views

Which polynomials make $\mathbb{R}$ into a monoid?

Question. Can we describe the set $$\{P : \mathbb{R}^2 \rightarrow \mathbb{R}, e:\mathbb{R}\mid P\; \mbox{is a bivariate polynomial}, P \mbox{ is associative}, P(e,x) = P(x,e) = x\}$$ explicitly? ...
3
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0answers
34 views

Example of a complex manifold with certain curvature properties?

Are there (nice) examples of a complex manifold such that the sectional curvatures through all the complex planes are non-positive but the sectional curvatures through the real planes are mixed?
7
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1answer
214 views

Find $f(x,y )$ such that $f_{x},f_{y},f_{yx}$ are continuous,but $f_{xy}$ is not

Let $f$ be a function of two variables,let$(a,b)$ be a point and let $D$ be an open disk with center $(a,b)$.Assume that $f$ is $\mathcal C^{1}$ on $D$,and $f_{yx}$ exist on $D$.Further ,the mixed ...
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0answers
18 views

Minimum edge cover of the Petersen graph

I want to find a minimum edge-cover of the Petersen graph. It is my understand that an edge cover is a set of vertices which is connected to all edges in the graph. Is this correct? I'm struggling to ...
3
votes
1answer
57 views

If $x$ is even and $y$ is odd, then $x+y$ is even.

I was also asked to proof if I say the above statement is true and give a counter example if I say it is False. Moreover, I prefer the statement to be false because the sum of any even and odd number ...
0
votes
2answers
38 views

Convolutions with $L^\infty$ functions

I read the following theorem about convolutions with $L^p$ functions in real analysis: Let ${\phi_n \in C^\infty_c({\bf R}^d)}$ be a sequence of approximations to the identity. If ${f \in L^p({\bf ...
4
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3answers
39 views

Rate of convergence of summable sequence

Suppose $a_n$ is a nonnegative real sequence such that \begin{equation} \sum_n a_n <\infty. \end{equation} What do we know about $a_n$? We know $a_n\to 0$. We know $$\lim\inf n a_n = 0.$$ But can ...
7
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2answers
620 views

Example of a continuous function with a discontinuous inverse

What is an example of a function $f: \Bbb R^n \rightarrow \Bbb R^m$ such that $f$ is continuous and injective but that $f^{-1}$ is not continuous. Our professor teased us with the notion but I ...
1
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1answer
31 views

Restriction of an isomorphism to an invariant subspace may fail to be surjective

I'm wondering whether the restriction of a vector space automorphism $f : V \to V$ to an invariant subspace $W \subset V$ can fail to be surjective, i.e. $f\vert_W : W \to W$ is not an ...
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2answers
68 views

Does $f(0) = 0 \implies f'(x) = 0$? [closed]

Assuming $f(x)$ is differentiable $\forall x$ In my textbook, for one of the questions, it says $f(0) = f'(0) = 0$, I was a little confused since I thought $f(0) = 0 \implies f'(x) = 0$ and thought ...
2
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2answers
62 views

An example of ideal $I$ such that $I^{ec}\neq I$

Let $A$ be a commutative ring, $S \subseteq A$ a multiplicative system and $i_S : A \rightarrow S^{-1}A$ the canonical morphism. Can you give me an example of ideal $I \unlhd A$ such that $I^{ec}\neq ...
0
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0answers
43 views

Linear Algebra, linear transformation

I was asked to give an example that satisfies the properties of a linear transformation, but that is not a function. Any help please? The question is from my teacher, and he insists that there ...
6
votes
1answer
57 views

Can $f_n\to f$ uniformly, $f'_n\to g$ uniformly, but $f$ not being differentiable?

Just the question in the title, I know that if $f_n$ are differentiable, $f_n\to f$ uniformly, $f'_n\to g$ uniformly and $f$ is differentiable, then $f'=g$, so I'm looking for a counterexample if we ...
4
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1answer
103 views

Two points of view on constructible sets

This question is aimed at understanding the relationship between two different definitions of the constructible sets in a Noetherian scheme, both of which I encountered in Atiyah-MacDonald's ...
2
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0answers
47 views

Showing that the field of rational functions is not dense

I am going through Counter Examples of Analysis but I am having trouble understanding a claim it makes. The book establishes that the set of rational functions defines an ordered field where the ...
-2
votes
2answers
70 views

'Non-example' to the Lagrange theorem. [duplicate]

Lagrange's theorem: Let G be a finite group and let H be a subgroup of G. Then, $|H| | |G|$ The converse does not hold in general. 'Non-example': $G=A_{4}$ where $A_{4}$ is the alternating ...
3
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1answer
39 views

Example for uniformly integrable $\mathbb{L}^2$-bounded sequences

How can I construct an example to show that the sequence $\{X_n\}_{n \ge 1}$ can be $\mathbb{L}^2$-bounded, however it has $\mathbb{E}\left[\sup\limits_{n\ge 1} |X_n|\right]=\infty$. Basically I need ...
1
vote
1answer
34 views

A uniformly integrable sequence that is not uniformly integrable in $\mathbb{L}^p$ for $p>1$

I need to construct an example to show that if for the sequence $\{X_n\}_{n \ge 1}$ we have that $\mathbb{E}\left[\sup\limits_{n \ge 1} |X_n|\right] < \infty$, the sequence in not necessarily ...
1
vote
1answer
31 views

Counterexample for uniform integrability of an $\mathbb{L}^1$-bounded sequence

I need to find an example such that $\sup\limits_{n \ge 1} \mathbb{E}[|X_n|] < \infty$ but $\{X_n\}_{n \ge 1}$ are not uniformly integrable. I can show that if $\{X_n\}_{n \ge 1}$ are uniformly ...
2
votes
4answers
44 views

Numerical property $P(n)$ such that $\forall n P(n)$ is false but a counterexample is difficult to find

I would like to find a nontrivial property $P(n)$ for $n \in \mathbb N$ such that $\forall n P(n)$ is false but the first counterexample can be found only for "very high" $n$ (so high that it wouldn't ...
1
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1answer
23 views

A positive square integrable random variable whit non square integrable inverse

I'm looking for an example of a Square Integrable Random Variable, whose multiplicative inverse is not Square Integrable.
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2answers
55 views

Unnatural homomorphism form domain $R$ to $Frac (R)$

There is a natural homomorphism for $R$ to $Frac (R)$ that sends $r\rightarrow(r,1)$, but beside this injective homomorphism, is there example of ring $R$ s.t there exist another injective ...
3
votes
1answer
79 views

If $\min(\alpha,F)$ has only one root in $E$, must $\min(p(\alpha),F)$ have only one root in $E$

Let $F<E$ be an algebraic field extension. Let $\alpha\in E$ be such that $\min(\alpha,F)$ has only one root in $E$ (which will be $\alpha$).Is it true that for any $p(x)\in F[x]$ we must have: ...
1
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0answers
29 views

Two alleged counterexamples (about boolean algebras)

Trying to solve this question, I propose two possible counter-examples. Please help me to understand whether these cases are really counter-examples. Let $\mathfrak{A}$ and $\mathfrak{B}$ be (fixed) ...
1
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1answer
52 views

Show that every nearly compact space is almost compact space but the converse is not true

I am learning about the almost compact space and nearly compact space. I know that every nearly compact space is almost compact space but the converse is not true in general. So i need an example of ...
10
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1answer
148 views

What are some interesting counterexamples given by finite topological spaces?

According to Wikipedia, 'finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures.' I have been studying the book ...
0
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1answer
24 views

Counter example of Algebraic sets

Affine n - spaces over a field $K$ is the cartesian product of the field $K$ with itself $n$ time and it is denoted by $\mathbb A^n(K)$. $X$ is a subset of Afine n - spaces $\mathbb A^n(K)$ is called ...
3
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1answer
50 views

Does equal cardinality and one set being a subset of the other prove equality? [closed]

I'm currently solving a quite specific problem and in the final step I made a statement that can be generalised such that: $$((|A|=|B|)\wedge(A\subset B)) \implies (A=B)$$ Whilst this is clearly ...