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.
5,866 questions
1 vote
1 answer
54 views
Function continuous nowhere whose domain and range are $[0,1]$
Find a function continuous nowhere, whose domain and range are both $[0,1]$. My intuition was to start with $f(x)=x$ and exchange to $f(a)=b$, $f(b)=a$ for sufficiently many pairs of $(a,b)$. So I ...
- 4,892
1 vote
1 answer
75 views
Can $\lim\limits_{(x,y)\to0}f(x)$ doesn't exist but at every path $y=\sum\limits_{k=1}^na_k x^k$, $\lim\limits_{(x,y)\to0}f(x)$ exist and are equal.
This is a generalization of this question A quick and easy was to prove that a 2 dimensional limit like $$\lim\limits_{(x,y)\to0}\frac{xy}{x^2+y^2}$$ is to try 2 different linear paths and prove that ...
- 9,055
7 votes
2 answers
396 views
Series that is known to converge/diverge but for which all these standard tests are inconclusive .
I have noticed that nearly every series I have been asked to analyze its convergence or divergence can be handled by the usual collection of tests: the limit test, Cauchy condensation, the integral ...
- 9,055
2 votes
2 answers
76 views
Minkowski sum of the set of irrational with an open interval is open?
"If $A$ and $B$ are subsets of the set of real numbers where either $A$ or $B$ is open, then the Minkowski sum of $A$ and $B$ is open". I am failing to see how it can be true, as any real ...
1 vote
1 answer
89 views
Conditions for Separable $T_{3\frac{1}{2}}$ Spaces to be Strongly Paracompact
It's known that $T_3$ Lindelof spaces are strongly paracompact, but I was wondering what sorts of conditions are needed to ensure strong paracompactness. For $T_3$ locally Lindelof spaces, ...
- 734
0 votes
2 answers
175 views
Counterexample to "the square of a non-monotone function is non-monotone"
The question: “If a function is not monotone on $(a, b)$, then its square cannot be monotone on $(a, b)$.” We are to provide a counterexample to this statement. On initial attempts I was able to forge ...
- 3
4 votes
1 answer
162 views
What properties does the one-point compactification of the Arens-Fort space have?
This is intended to be a self-answering question, which is allowed on StackExchange sites (see here). We are interested in the traits of the one-point compactification of the Arens-Fort space not yet ...
- 189
-2 votes
1 answer
87 views
If $(x_n)$ is a bounded sequence and $f$ is a continuous function, must $(f(x_n))$ be bounded? [closed]
If $(x_n)$ is a bounded sequence and $f$ is a continuous function, must $\bigl(f(x_n)\bigr)$ be bounded? I think not. For example, take $f(x) = \frac{1}{x}$, and $x_n = \frac{1}{n}$. Then $x_n$ is ...
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