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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294 views

Distance from a compact subset need not be attained in a metric space?

Suppose we have a metric space $(X,d)$. Let $S$ be a compact subset of $X$. Provide me with an example of $X$, and $S$ (closed and bounded in $X$) such that $$\min \{d(p_0,p): p \in S\}$$ does not ...
0
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2answers
21 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: ...
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2answers
50 views

example of a convergent series that $\lim \sup |\frac{z_{n+1}}{z_n} | > 1$

Let $(z_n) \subset \mathbb C$, with $z_n \neq 0$. It's known that if $\lim \sup |\frac{z_{n+1}}{z_n} | < 1$, so $\sum |z_n|$ converges, then $\sum z_n$ converges. Can I find an example of a ...
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0answers
11 views

An example that does not satisfy the conditions of the Fourier inversion theorem?

Here is the Fourier inversion theorem page in Wikipedia. It states that for every function $f(x)$ that satisfy some conditions ($f(x)$ can be a function such as a Schwartz function, an integrable ...
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1answer
35 views

Essential singularity of the resolvent operator of an unbounded operator

Is there an unbounded operator with isolated points in the spectrum, not all of which are eigenvalues? For unbounded operators it is known that isolated spectral points are either poles or essential ...
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2answers
26 views

Series that converges on $[-1,1]$ [on hold]

What is an example of a series that converges only on $[-1,1]$? I am unable to come up with one right now for some reason. Thanks
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2answers
82 views

Examples of Manifolds such that $\chi (X)=-3$

I am trying to come up with an example of a closed oriented manifold with euler characteristic equal to $-3$. I have tried to use $\chi (\underbrace{T^2\mathbin{\#}\cdots \mathbin{\#} T^2}_{\text{$g$ ...
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1answer
76 views

a dense set in plane

Is there a dense set in $\Bbb{R^2}$ that every vertical line or horizontal line intersect in finite points. I think that we can consider $\Bbb{Q} ×\Bbb{Q}$ but every vertical line or horizontal line ...
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1answer
27 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 ...
4
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1answer
197 views

The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
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6answers
500 views

If $G$ is a group and $N$ is a nontrivial normal subgroup, can $G/N \cong G$? [duplicate]

I know $G/N$ is isomorphic to a proper subgroup of $G$ in this case, so the gut instinct I had was 'no'. But there are examples of groups that are isomorphic to proper subgroups, such as the integers ...
3
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2answers
184 views

finding a function with predetermined $f^{(n)}$s at $0$ and $1$.

Let $a_n,b_n$ be 2 arbitrary sequences of real numbers. Is there any $C^\infty$ function $f:\mathbb{R}\to\mathbb{R}$ such that for any $n$: $$f^{(n)}(0)=a_n,\quad f^{(n)}(1)=b_n$$
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2answers
220 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
18 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 ...
3
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1answer
30 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 ...
7
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0answers
58 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
43 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 ...
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1answer
45 views

What kind of functions lie in $L^2(\Omega )-H^{1/2}(\Omega )$?

It's well known that a discontinuous function does not lie in $H^{1/2}(\Omega )$ if $\Omega\subset\mathbb{R}$ is one dimension. My question is, For $\Omega\subset \mathbb{R}^3$, are there any simple ...
1
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0answers
16 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 ...
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, ...
0
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1answer
54 views

Can a complex function be complex-differentiable at a point and not in a neighborhood?

Is it possible for a function $f:\mathbb{C} \to \mathbb{C}$ to be complex-differentiable at a point $z_0\in \mathbb{C}$ without being analytic in a neighborhood of $z_0$? How can we prove this?
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3answers
612 views

Papers with unorthodox writing style

I'm not sure if this is the right forum for this question, in any case probably CW is appropriate? I've been looking around the mathblogosphere for the past few weeks and ran into mathgen. It's ...
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 ...
6
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3answers
2k views

An ideal that is radical but not prime.

I'm preparing for an exam and, as part of this preparation, I'm looking for an ideal $I$ in an integral domain $R$ that is radical but not prime. Here is an example I'm fooling around with: ...
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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 ...
6
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2answers
788 views

Spaces with equal homotopy groups but different homology groups?

Since it's fairly easy to come up with a two spaces that have different homotopy groups but the same homology groups ($S^2\times S^4$ and $\mathbb{C}\textrm{P}^3$). Are there any nice examples of ...
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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
26 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 ...
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3answers
3k views

Continuous and bounded variation does not imply absolutely continuous

I know that a continuous function which is a BV may not be absolutely continuous. Is there an example of such a function? I was looking for a BV whose derivative is not Lebesgue integrable but I ...
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4answers
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Real life examples of commutative but non-associative operations

I've been trying to find ways to explain to people why associativity is important. Subtraction is a good example of something that isn't associative, but it is not commutative. So the best I could ...
9
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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 ...
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 ...
2
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1answer
45 views

Is a ring $R$ factorial $\iff$ $R[X]$ factorial?

Let $R$ be a factorial ring. Then, the polynomial ring $R[X]$ is factorial. I was wondering if the other direction also works (i.e. $R[X]$ factorial $\implies$ $R$ factorial)? If not, please give ...
0
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1answer
76 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
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0answers
21 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. ...
4
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1answer
366 views

Example of a function that has the Luzin $n$-property and is not absolutely continuous.

The Banach–Zaretsky theorem (page 196) says that a continuous function $f:[a,b]\to\mathbb{R}$ of bounded variation is absolutely continuous if and only if $$E\subset I \text{ has zero Lebesgue ...
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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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?
7
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2answers
2k views

Examples of prime ideals that are not maximal

I would like to know of some examples of a prime ideal that is not maximal in some commutative ring with unity.
142
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0answers
3k 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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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 ...
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2answers
36 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
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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 ...
3
votes
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? ...
5
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5answers
131 views

Convergent sequence of irrational numbers that has a rational limit.

Is it possible to have a convergent sequence whose terms are all irrational but whose limit is rational?
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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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3answers
70 views

Is there a countably compact sequential non-$T_2$ space that is not sequentially compact?

Let $X$ be a topological space. Definitions: $X$ is countably compact if every countable open cover of $X$ has a finite subcover or equivalently, every sequence in $X$ has a cluster point. $X$ is ...
7
votes
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 ...