Questions tagged [examples-counterexamples]

To be used for questions whose central topic is a request for examples where a mathematical property holds, or counterexamples where it does not hold. This tag should be used in conjunction with another tag to clearly specify the subject.

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31 views

Abel's test for improper integrals with only integrable functions

I know this following formulation of Abel's test for improper integrals: Let $f,g:[a,\infty)\to \mathbb{R}$ be continuous functions, where $\int_a^\infty f(t)dt$ converges. $g$ is monotone ...
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1answer
40 views

Example of a function whose second derivative does not exist but limiting formula for the second derivative holds

Here's Exercise 11 in Baby Rudin: Suppose $f$ is defined in a neighborhood of $x$, and suppose $f^{\prime\prime}(x)$ exists. Show that \begin{equation}\label{11.0} \lim_{h \to 0} \frac{f(x+h)+ f(...
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28 views

Localization of a infinite sum of ideals not equal to infinite sum of localized ideals

I'm trying to find a counterexample to $$ S^{-1} \left(\displaystyle\sum_{i=1}^\infty I_i \right) = \displaystyle\sum_{i=1}^\infty S^{-1}I_i, $$ where the $I_i$ are ideals on a commutative ring $A$ ...
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10 people formed a team . The team is always at least 3 people. Every evening, 1 person is added to the team or 1 person is excluded from it. [closed]

10 circle members formed an on-duty team to solve homework. The team is always at least 3 people. Every evening, one person is added to the team or one person is excluded from it. Will it be ...
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1answer
28 views

Does every non-compact Tychonoff space admit an unbounded continuous function? [duplicate]

Let $X$ be a completely regular Hausdorff space. Such a space is also known as Tychonoff space, or a $T_{3.5}$-space. Furthermore, let's assume that $X$ is not compact. Question. Does $X$ admit a ...
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2answers
108 views

Are the solutions of $f(x+h)=f(x)f(h)$of the form $a^x$ even if we consider not continuous functions

Let $$f(x):\mathbb{R}\to \mathbb{R} $$$$$$and$$f(x+h)=f(x)f(h)$$ If $f(x)$ is a continuous function then we can prove all solutions for ($f(x)$ not equal to zero at any point) are of the form $a^x$ ...
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Would a finite number of primes imply there would be a finite number of secure exchanges on the internet?

I was teaching my class Euclid's theorem on why there are infinite number of primes. Aside from the idea of proof by contradiction, I wanted to give some more motivation as to why knowing this fact is ...
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76 views

Are $\pi(y)$ and $\pi(xy)$ associates?

Let us assume a commutative ring $A = \mathbb{R}[x, y]/(yx^2 - y)$ and a natural homomorphism $\pi: \mathbb{R}[x, y] \rightarrow A = \mathbb{R}[x, y]/(yx^2 - y)$ that sends $p(x, y)$ to $p(x, y) + (yx^...
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72 views

Does convexity at a single point imply convexity w.r.t finite convex combinations?

Let $\phi:\mathbb (0,\infty) \to [0,\infty)$ be a continuous function, and let $c \in (0,\infty)$ be fixed. Suppose that "$\phi$ is convex at $c$". i.e. for any $x_1,x_2>0, \alpha \in [0,...
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74 views

Does convexity at a point imply existence of one-sided derivatives?

Let $\phi:\mathbb (0,\infty) \to [0,\infty)$ be a continuous function, and let $c \in (0,\infty)$ be fixed. Suppose that "$\phi$ is convex at $c$". i.e. for any $x_1,x_2>0, \alpha \in [0,...
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1answer
65 views

Are there invertible functions such that $f=\frac{g}{h}$ and $f^{-1}=\frac{g^{-1}}{h^{-1}}$?

Note: this question is inspired by this one: Why $\arctan x$ not equal to $\arcsin(x)/\arccos(x)$? In the linked question, it is said that if $f=\frac{g}{h}$ and $f$, $g$ and $h$ are invertible, then $...
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1answer
76 views

If continuous images of $X$ are closed in every $Y$, is $X$ a compact space?

Suppose $X$ is a topological space. We have the following criterion for compactness: Theorem. $X$ is compact if and only if for every space $Y$, the second projection $\pi_2: X\times Y \to Y$ is a ...
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2answers
59 views

Help needed in Assignment questions in Complex Analysis( True - False Questions ) [closed]

I am trying assignments of complex analysis and I am unable to solve the some questions which I am asking here as I am really confused on how they can be approached . Questions : State whether true ...
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2answers
57 views

Understanding the substitution theorem of Riemann integration.

Let us say $f$ is an integrable function on $[a,b]$ and we want to evaluate $\int_a^b f(x)dx$ but often the calculation is not easy.So,we have a method of substitution.We substitute $x=\phi(t)$ where $...
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1answer
33 views

Example / Counterexample of non constant analytic function

While trying assignments of complex analysis I am unable to solve this particular question. Does there exists a non-constant bounded analytic function on $\mathbb{C} $/{0} ? As function is not ...
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2answers
67 views

Does $\langle f(x),x\rangle=\langle g(x),x\rangle$ imply that $f=g$? [duplicate]

Let $V$ be a a Euclidean or unitary vector space and $f,g$ be linear endomorphisms from $V$ such that $\langle f(x),x\rangle=\langle g(x),x\rangle,\,\forall x\in V$ holds. Does $f=g$ hold? Does it ...
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1answer
34 views

Examples of singular non locally constant functions

What are (as easy as possible) examples of functions $f$ with the following properties? singular, i.e. continuous, non-constant, and differentiable almost everywhere with derivative zero, non locally ...
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1answer
19 views

Find a counter-expamle to $\lim_n \limsup_m d(a_n, a_m) =0 \implies (a_n)_n \ \text{is cauchy}.$

Let $(a_n)$ be a sequence in a $d$-metric space. I want to find a counter-example to the statement $$\lim_n \limsup_m d(a_n, a_m) =0 \implies (a_n) \ \text{is Cauchy}.$$ I know that $\lim_n \limsup_m ...
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0answers
18 views

Maximum bottleneck-capacity variant of Ford-Fulkerson: worst-case example

I am looking for a worst-case example (or series of examples) for the maximum bottleneck-capacity variant of Ford-Fulkerson (i.e. the path with the highest bottleneck-capacity is chosen as an ...
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2answers
80 views

Is it true that only one root results in a valid solution?

This question is a follow up on the question about construction the triangle given distinct altitude, bisector and median. The answer provides an expression for the side length $a$ as the two roots of ...
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1answer
91 views

Does this partial midpoint-convexity imply full convexity?

Let $f:(-\infty,0] \to \mathbb [0,\infty)$ be a $C^1$ strictly decreasing function. Definiton: Given $c \in (-\infty,0]$, we say that $f$ is midpoint-convex at the point $c$ if $$ f((x+y)/2) \le (f(x) ...
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1answer
47 views

“Natural” example of an undecidable subset of $\Bbb N$

All the simple examples of undecidable problems that I know deal with symbolic computation or calculation. For example, the halting problem, whether Diophantine equations have solutions, the word ...
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1answer
22 views

Function composition and inflection points [closed]

Considering two functions in $\mathbb{R}$ , $f$ and $g$, both having an inflection point on the same x-coordinate, does the function $h=f \circ g$ necessarily have an inflection point on that x-...
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1answer
27 views

If $\frac{a}{k} - \frac{a(c-1)}{kb} > 1$, is $\frac{a}{k+1} - \frac{a(c-1)}{(k+1)b} > 1$?

I'm trying an induction method, but not sure if I'm able to prove it or find a counterexample. Suppose $3 \leq a < b$ and $1 \leq c \leq b$, and $k \geq 1$, where $a, b, c, k$ are integers. Suppose ...
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37 views

Examples of Wishart matrices

I know the following definition: Matrix $A \in R^{n \times n}$ follows the Wishart distribution, i.e $A \sim W_n(m,\Sigma)$ when $$A=\sum_{j=1}^m X_jX_j^T=\begin{pmatrix}A_{11}&A_{12}\\A_{21}&...
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20 views

Is there an improper subset that isn't equal to its superset?

Can there be a set $A$ and a set $B$ such that $A\subseteq B$ and $A\ne B$ ? While trying to find a solution to this question, I've found this answer which states: An improper subset (usually denoted ...
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27 views

Counter-example to $\nabla^2 f =0 \implies \nabla \times \nabla f =\vec 0$?

I'm trying to answer the following question: "Let $f: \mathbb{R}^3 \to \mathbb{R}$ be a function satisfying $\nabla^2 f =0$, then $\nabla \cdot \nabla f =0$ and $\nabla \times \nabla f =\vec 0$&...
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30 views

Example counterexamples in a question on homeomorphism [closed]

This true false question was asked in my quiz yesterday but I am not sure about my answer for the same. State True or False : If f : [0,1] -> [-π,π] is a continuous bijection then it's a ...
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2answers
66 views

Slow variation of counting functions

Let $A \subseteq \mathbb{N} = \{1, 2, 3, \dots\}$ and define its counting function $\mathbb{R} \to \mathbb{R}$ to be $$A(x) = \#\{a \in A : a \leqslant x\}.$$ If $A \ne \varnothing$ then is it true ...
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1answer
45 views

Integration with respect to Borel measure

Context first: I'm brushing up a little on measure theory, mainly trying to get a better understanding of integration with respect to measures other than the Lebesgue measure. I came across the ...
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29 views

Which is an example of a monoidal category which cannot be braided?

This is probably a most stupid question, but I really do not have a profound knowledge of monoidal and monoidal braided categories, I only skimmed across them in a course on Hopf Algebras. My question ...
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1answer
38 views

Is the composition of closed operators closed?

Let $K,G,H$ be Hilbert spaces with $D_A \subseteq K$, $D_B \subseteq G$ (possibly not dense) subspaces and let $A: D_A \rightarrow K$ and $B:D_B \rightarrow H$ be closed linear operators. Then is the ...
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56 views

Can I say that every integrable function is measurable?

Let $(X, \mathcal S, \mu)$ be a measure space. Let $\Bbb L$ be the collection of all $\mathcal S$-measurable functions and let $L_1(\mu)$ be the collection of all $\mu$-integrable functions i.e. the ...
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27 views

What causes the solution(s) of different problems to be the same?

I've been wondering recently about this situation: If we have two or more different problems and their solutions turn out to be the same, what causes this phenomenon? Allow me to give some context. I'...
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1answer
36 views

Nontrivial example of a ring in which the union of ideals is an ideal

If R is a field and it has two ideal and any union of two ideal is again ideal. But Can we give an example of commutative ring(necessarily not field) with 1 and union of two ideal is again ideal?
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1answer
26 views

Space which is path connected but not locally path connected [duplicate]

Can you give an example of a topological space which is path connected but not locally path connected, besides the graph of $\sin(1/x)$?
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2answers
71 views

Are there examples of continuous, non-differentiable functions whose “rational derivative” exists?

Define the operator $\Delta_n$ according to the equation $$\Delta_nf(x)=f\left(x+\frac1n\right)-f(x)$$ Observe that for differentiable $f:\Bbb{R}\to\Bbb{R}$ $$\frac{df}{dx}=\lim_{n\to\infty}n\Delta_nf$...
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1answer
46 views

Closed set that is not complete

Can anyone help me out with an example of a closed set that is not complete? I have read up on the set of irrational numbers with the euclidean metric is such an example on other web pages, but that ...
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2answers
117 views

How can I construct a nilpotent matrix of order 100 and index 98?

I know to construct a nilpotent matrix of order $n$ with index of nilpotency $n$, but how to construct a nilpotent matrix of order $n$ but index of nilpotency $(n-2)$? Is there any general rule for ...
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2answers
241 views

Showing that Lebesgue Dominated convergence theorem is false in case of Riemann integration.

I was reading Tom Apostol book called "Mathematical Analysis" and I read this statement: the Lebesgue Dominated convergence theorem is false in case of Riemann integration. Here is the ...
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9 views

The protosheaf $(\mathbb R, p, S_1 )$ is not an etale-sheaf.

The protosheaf $(\mathbb{R}, p, S_1 )$, where $p: \mathbb R → S_1$ is the local homeomor- phism given by $x\mapsto e^{2πix}$ , is not an etale-sheaf of abelian groups (for its stalks are not abelian ...
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35 views

Find an algebraic division algebra that is not finite dimensional

I want to find an algebraic division algebra that is not finite dimensional, but i don't want to do it in terms of field extensions nor anything like that. Instead of that, what i want to do is to ...
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1answer
10 views

Show that a sequence of PDF of normal distribution with running mean and unit variance is not bounded by an integrable function

I am trying to show that the condition of bounded by an integrable function is crucial in the Dominated Convergence Theorem. Consider a sequence of functions $(f_n)$ on $\mathbb{R}, $ which is ...
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68 views

What is a simple example of a reduced, noetherian, local ring of dimension $0$ which is not Gorenstein?

As the title says, I am looking for a noetherian local ring $R$ of dimension 0 which is reduced (and thus Cohen-Macaulay) but not Gorenstein. Due to Bruns, Herzog $-$ Cohen-Macaulay Rings Theorem 3.2....
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Example of objects and morphisms in h-TOP

I've started studying algebraic topology and I've come up with my first example of real category: the h-TOP category. In order to understand it better, I've thought about an example, and I'd like to ...
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1answer
74 views

Non-Examples of Functors and Categories

I'm preparing to deliver some lectures on homological algebra and category theory, and have found lots of nice long lists of examples of functors and categories arising in every-day mathematical ...
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38 views

Example of continuous function whose Newtonian potential is not twice differentiable.

I was reading Gilbarg & Trudinger and in there is a statement that the Newtonian potential of a continuous function need not be twice differentiable. I would appreciate it if someone could provide ...
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1answer
22 views

A question on order-isomorphism with $\mathbb N$.

Let $A$ be a countable subset of $\mathbb R$ which is well ordered with respect to usual ordering $\leq$ of $\mathbb R$.Then does $A$ have an order preserving bijection with a subset of $\mathbb N$? ...
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103 views

Goldbach twin and cousin primes

Good day. My question is a counterexample for the following: Is every even number greater than 4 the sum of a number that belongs to set the cousin primes with another number that belongs to the set ...
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6answers
678 views

Functions $f: \mathbb{Z}^{+}\to \mathbb{R}$ satisfying $x f(y) + y f(x) = (x+y) f(x^2+y^2)$

Let $\mathbb{Z}^{+}=\{1, 2, 3, ...\}$ denote the set of positive integers. Problem 1. Are there any non-constant functions $f\colon \mathbb{Z}^{+}\to\mathbb{R}$ such that $$ x f(y) + y f(x) = (x+y) f(...

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