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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Find a function $f :\Bbb R^2\to \Bbb R$ which is not differentiable at $(0,0)$ even though all directional derivatives of $f$ exist at $(0,0).$ [duplicate]

This is a practice exam question and I have no idea how to start it. Find a function $f :\Bbb R^2\to \Bbb R$ which is not differentiable at $(0,0)$ even though all directional derivatives of $f$ ...
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42 views

What problems are related with the following type of FDE with delay?

Consider the following class of functional differential equations with delay: $$\begin{align} \frac{du}{dt} &= F(x,t,u(x,t),u_{t,x}), & (x,t) &\in [a,b] \times [0,T] \\ u(x,t) &= ...
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find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges

Find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges. To make it more challenging, produce examples where $a_n$ and $b_n$ are ...
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1answer
25 views

How do I answer questions that ask to construct examples with some given properties?

Give examples of sets and functions with the following properties: an open set $D_1 \subseteq \mathbb{R}$ and a continous function $f:D_1\rightarrow \mathbb{R}$ such that $f$ has both a maximum and ...
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75 views

If $f_{xy}$ , $f_{yx}$ are continuous at $(x_{0},y_{0})$,then $f_{x},f_{y}$ are continuous at $(x_{0},y_{0})$?

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ The second edition Let $f$ be a function of two variables,let$(x_{0},y_{0})$ be a point and let $U$ be an open disk with center $(x_{0},y_{0})$.Assume ...
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difficult example of a not differentiable function $f: \mathbb R^2 \to \mathbb R^2$ [closed]

Give an example of a function $f: \mathbb R^2 \to \mathbb R^2$ so that: 1) all its directional derivatives exist at $(0,0)$ ($D_{\vec u}f(0,0)$ exist for all $\vec u \in \mathbb R^2$ unitary), 2) ...
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2answers
44 views

Finite Space that is Not Normal

Is there any finite space that is not normal? By "normal", I refer to a space in which disjoint closed sets can be separated by disjoint open sets.
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4answers
95 views

Find a linear operator s.t. $A^{2}=A^{3}$ but $A^{2}\neq A$?

From Halmos's Finite-Dimensional Vector Spaces, question 6a section 43, the section after projections. Find a linear transformation A such that $A^{2}(1-A)=0$ but A is not idempotent (I remember A is ...
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5answers
267 views

Is it true that $2^n-1$ is prime whenever $n$ is prime?

In my discrete math book, I was tasked with finding a counterexample for this: If $n$ is prime, then $2^n-1$ is prime. Does there exist a counterexample for such a statement? Also, am I wrong in ...
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Double mapping cylinder- a point set question

We have the following set up: $X_0 \subset X_{\pm} \subset X$. Also interiors of $X_\pm$ cover $X$. Now let $Z$ be the double mapping cylinder of the maps $X_- \leftarrow X_0 \rightarrow X_+$. Define ...
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20 views

Nonlinear Programming examples

I'm an optimization newbie. I am looking for a simple nonlinear optimization problem that I can work through in Excel. For linear optimization I used the "Giappeto Inc" problem and I wonder if there ...
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2answers
51 views

Explaining roundoff error when row reducing matrices

In my linear algebra textbook (in the context of row reducing and obtaining a matrix in echelon or reduced echelon form), there is a numerical note that reads as follows: "A computer program usually ...
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0answers
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Embedding vs embedding as a closed subset

A version of Whitney's Theorem state that any $n-$dimensional manifold can be embedded in $\mathbb{R}^{2n+1}$ as a closed subset. Another version states that that any $n-$dimensional manifold can be ...
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1answer
45 views

Example of non-commutative ring without unit such that…

Give an example of a non-commutative ring without unity such that $(xy)^2=x^2y^2$, for all $x,y\in R$.
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1answer
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Counterexample Math Books

I have been able to find several counterexample books in some math areas. For example: $\bullet$ Counterexamples in Analysis, Bernard R. Gelbaum, John M. H. Olmsted $\bullet$ Counterexamples in ...
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3answers
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What is an example of UFD such that a gcd of an infinite set does not exist?

Let $R$ be a unique factorization domain and $S$ be an infinite subset of $R$. What would be an example of $R$, $S$ such that a gcd of $S$ does not exist in $R$? That is, is there an infinite set ...
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1answer
48 views

Smooth function whose $(n+1)$th derivative is defined only on a proper subset of the domain of the $n$th, and the radius contract to $0$

I'm wondering if there exists such a function, whose $(n+1)$th derivative is defined only on a proper subset of the domain where the nth derivative is defined, and with the property that the diameter ...
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1answer
54 views

A compact infinite topological group with only two closed subgroups

It can be proved every compact infinite abelian topological group $(A,\tau )$, with $\tau$ nontrivial, has at least three distinct closed subgroups. Is there any compact infinite non-abelian ...
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1answer
70 views

Example of non-differentiable continuous function with all partial derivatives well defined

Give an example of a function $f : \mathbb{R}^3 \to \mathbb{R}$ such that the partial derivatives exist at $(0,0,0)$, and $f$ is continuous at $(0,0,0)$, but it is not differentiable at $(0,0,0)$. Any ...
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11answers
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Problems that become easier in a more general form

When solving a problem, we often look at some special cases first, then try to work our way up to the general case. It would be interesting to see some counterexamples to this mental process, i.e. ...
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2answers
535 views

Does there exist a continuous and differentiable function which isn't smooth?

As I understand, a smooth function is continuously differentiable. But if I have a function which is continuous AND differentiable, I cannot automatically say that it is smooth. For it has to be so ...
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1answer
65 views

Caratheodory: Inner vs. Outer

Problem Given a plain space $\Omega$ and a ring $\mathcal{R}$. (In fact, a semiring would do the job, too.) Consider a premeasure $\mu:\mathcal{R}\to\overline{\mathbb{R}}_+$. For simplicity, ...
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1answer
99 views

Differentiable function with a set of critical points of second category.

I'm looking for an example of a nowhere constant differentiable function with a set of critical points of second category. In other words: Let $U \subset \mathbb{R}$ open. Is there a differentiable ...
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1answer
34 views

Find a sequence $(x_n)$ such that $(x_n)$ is monotonic, $\lim x_n=0$, $\sum_{n=1}^{\infty} ( 1-\frac{x_n}{x_{n+1}})$ converges

Find a sequence $(x_n)$ satisfying these conditions: $(x_n)$ is monotonic and $\lim x_n=0$ $\displaystyle \sum_{n=1}^{\infty} \left( 1-\frac{x_n}{x_{n+1}}\right)$ converges. This ...
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1answer
40 views

A compact Hausdorff space that is not Tychonoff

Is there an example of compact Haudorff that is not Tychonoff? As every continuous function on compact space is bounded, then I was thinking maybe every compact Haudorff is Tychonoff but I failed to ...
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1answer
44 views

A pushout of a homotopy equivalence along

Can anybody show me an example which prove that: A pushout of a homotopy equivalence along a arbitrary map (in Top) doesn't have to be a homotopy equivalence. I know that if we change "arbitrary ...
2
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1answer
44 views

Isomorphic relation between dihedral groups

Theorem Let $G,H$ be abelian groups such that $Dih(G)\cong Dih(H)$ If $G$ is finitely generated, then $G\cong H$. I'm curious whether "finitely generated" hypothesis can be removed. If ...
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1answer
21 views

Example of H-set

A subset $Z$ of a topological space $X$ is said to be an $\textbf{H-set}$ if there exists a transfinite decreasing sequence $\{ F_{\sigma}:\alpha < \kappa \}$ of closed subsets of $X$ such that ...
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1answer
39 views

$X$ is compact Hausdorff iff it is pseudocompact and realcompact

I just read this article http://en.wikipedia.org/wiki/Realcompact_space. I am interested with a property: $X$ is compact Hausdorff iff it is pseudocompact and realcompact. I don't know how to prove ...
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1answer
82 views

Finding a function ,satisfying the given properties

Finding a function of two variables , satisfying $$\lim_{\left(x,y \right)\rightarrow \left({x}_{0},{y}_{0} \right)}f\left(x,y \right)=+\infty ;$$ and for $\forall\delta>0, \exists y^{'},y^{''}\in ...
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1answer
368 views

What are the requirements for separability inheritance

Suppose we have an arbitrary separable topological space $X$. What are some (possibly nonequivalent) minimal requirements to put on $X$ to ensure that every subspace of $X$ is separable? This is not ...
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2answers
78 views

Relation between chief and compositions series of a group

Is there an example of a group with a composition series (of finite length) but without a chief series (of finite length)? Is there an example of a group with a chief series (of finite length) but ...
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1answer
102 views

Developing Examples for Basis $\mathcal{B}$ on Topology $\tau$

I put together my own example on the basis of topology. I wanted to know if it is a valid example displaying the properties of basis. Here is my example: Example: Since $X\subset\mathbb{R}$, let ...
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28 views

Example of infinite sequences that produces different norms

So I have an infinite sequence: $u = (u_1, u_2, u_3, . . .)$, where $u_j$ are real numbers. $l_1, l_2, l_\infty$ is defined as $$||u||_1=\sum_{j\in \mathbb{N}} |u_j|$$ ...
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77 views

Examples (trivial and non-trivial) of computable functions whose inverse is not computable

Can you give some examples (some trivial and some non-trivial) of computable functions whose inverse is not computable?
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2answers
35 views

Why does $f(U)$ is open for every open $U\subset M$ not imply $f$ is continuous?

Let $f:M \to N$ be a map from a metric space $M$ to a metric space $N$. Does "$f(U)$ is open for every open $U\subset M$" imply $f$ is continuous? I think it's wrong but I can't find a counter ...
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Non-physical Jounce Examples in Nature

What are some good examples of jounce, the fourth derivative of position, in the non-physics arena? The reason I ask is that A) it's already difficult for a lay to visualize it in the physical arena ...
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2answers
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Counterexample: Nonvanishing derivative (constant) on $\Bbb R$ implies injectivity?

I know the case is false if $S \subset \Bbb R$ by Rolle's Theorem, what is a counter example if $S = \Bbb R$? I think the "analog" is false on $\Bbb C$, but that's yet to be proven? Addedeum: ...
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2answers
66 views

Why must a locally compact second countable Hausdorff space be second countable to imply paracompactness?

The textbook version of the result I've seen states: A locally compact second countable Hausdorff space is paracompact. Is the property of being second countable needed, or have I missed something? ...
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1answer
61 views

Strongly mixing uniquely ergodic dynamical system

I'm looking for an example of a dynamical system which is both (measure-theoretically) strongly mixing and uniquely ergodic. I've looked around and found lots of discussion of systems which are ...
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2answers
67 views

give an example of a cyclic group with 6 generators.

Give an example of a cyclic group with 6 generators. Give the generators, explain how you know that these are generators and that they are the only generators. I don't even know how to begin this ...
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1answer
50 views

A nonmetrizable image of a metrizable space

It is well known that a hausdorff continuous image of a compact metric space is metrizable. What is a counterexample for noncompact case?
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An example of a nilpotent group

Is there an example of a nilpotent group such that $G/G'$ is (non-trivial) torsion-free while $G$ is not? I cannot think of any example of this kind and I think that it is not proved any result like ...
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2answers
443 views

Injectivity of Homomorphism in Localization

Let $\alpha:A\to B$ be a ring homomorphism, $Q\subset B$ a prime ideal, $P=\alpha^{-1}(Q)\subset A$ a prime ideal. Consider the natural map $\alpha_Q:A_P\to B_Q$ defined by ...
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4answers
313 views

give a counterexample of monoid

If $G$ is a monoid, $e$ is its identity, if $ab=e$ and $ac=e$, can you give me a counterexample such that $b\neq c$? If not, please prove $b=c$. Thanks a lot.
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Both $F$ and $C$ are closed sets but their sum $F+C$ is not closed. [duplicate]

In context to the question what will be an counter example such that both $F$ and $C$ are closed sets in $ \Bbb R^n$ but their sum $F+C$ is not closed in $ \Bbb R^n$?
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Primes of the form $\frac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $n,a,b$ be positive integers with $a<b$. Consider primes of the form $f(n)=\dfrac{n^2-n+4}{2}$. Let $C(a,b)$ denote the amount of primes of the form $f(n)$ between (and including) $f(a)$ and ...
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1answer
26 views

Metric spaces - Limit points and Isolated points

I apologise for this, but I have numerous potential misunderstandings. $\Bbb R^2$ is a metric space, since it is a subspace of the Euclidean space, $\Bbb R^n$ I can look at a set $E$ with elements ...
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2answers
51 views

Laplace transform convolution attempt

I can't seem to get this Laplace working using the convolution method. $H(s) = \frac{1}{s^2(s+2)}$ Which I can't get to work using convolution. So I am separating it into $\frac{1}{s^2} * ...
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1answer
24 views

Difference between finite and discrete set of discontinuities

I am reading this paper. In page 395 and 396, theorems $1$ and $2$ use the terms 'finite' and 'discrete' to refer to sets, in this case sets of discontinuities. What I don't understand is: what is the ...