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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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 $n$: $$f^{(n)}(0)=a_n,\quad f^{(n)}(1)=b_n$$
1
vote
2answers
239 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.
0
votes
1answer
20 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
votes
1answer
50 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
votes
0answers
64 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 ...
1
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0answers
48 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
vote
1answer
49 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
21 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, $$f(z)-f(a)...
0
votes
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?
15
votes
3answers
633 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
vote
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
votes
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: ...
2
votes
2answers
39 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
votes
2answers
834 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 ...
0
votes
0answers
15 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
votes
1answer
38 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 ...
10
votes
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 ...
23
votes
4answers
6k views

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
votes
3answers
5k views

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
vote
1answer
34 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
votes
3answers
42 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 $\bigcap_{...
2
votes
1answer
53 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
votes
1answer
83 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
23 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
votes
1answer
420 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
votes
0answers
30 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
votes
2answers
42 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
votes
2answers
3k 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.
159
votes
0answers
4k 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, $f(S)\subset Y$ ...
0
votes
0answers
19 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) = f(...
1
vote
1answer
41 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 idea: ...
0
votes
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
-1
votes
2answers
74 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
61 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
votes
5answers
133 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?
1
vote
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
votes
3answers
73 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 ...
18
votes
2answers
6k views

Intersection of finite number of compact sets is compact?

Is the the intersection of a finite number of compact sets is compact? If not please give a counter example to demonstrate this is not true. I said that this is true because the intersection of ...
3
votes
0answers
38 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?
2
votes
3answers
1k views

Proof by contradiction using counterexample

Why can't we use one counterexample as the contradiction to the contradicting statement? Example: Let a statement be A where a-->b. We can prove A is not true by finding a counter example. Now, in ...
1
vote
0answers
20 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 ...
42
votes
10answers
13k views

Examples of finite nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
2
votes
1answer
61 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 ...
3
votes
1answer
63 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 ...
3
votes
1answer
142 views

Give example of a series $\sum a_n$ such that the series is conditionally convergent. and $\sum na_n$ is convergent

I tried all the conditionally convergent series I know, I found $\sum na_n$ to be diverging for all of them. But I am sure the question is correct
3
votes
1answer
288 views

$\sum a_n$ divergent with $\lim(n a_n)=0$

Can we find an example of a divergent series $$\sum_{n=1}^\infty a_n$$ where the sequence $(a_n)$ is a decreasing sequence of real numbers, but such that $$\lim_{n\to\infty}(n a_n)=0$$
1
vote
1answer
158 views

Modifications of the necessary condition for series convergence [closed]

Could you help me prove the two following lemmas on series convergence? 1) Prove that if $\sum_{n=1} ^{\infty} a_n$ is a series of positive real numbers convergent to $0$, where $(a_n)$ is a monotone ...
4
votes
3answers
43 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 ...