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