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.
5,870 questions
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Matrices that are related to their transpose via diagonalizable matrices
I'm currently working with matrices having the following property:
Let $A \in M_n(\mathbb Z)$ be square matrix such that there exist diagonalizable matrices $S,T \in M_n(\mathbb C)$ with $A = S A^t T$,...
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Counter example to dominated convergence for nets
I came across the following counter example in the accepted answer to this questions A net version of dominated convergence?
For context, the original example is this:
Let $\Lambda$ be the set of ...
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Counterexample for Unitary Operators and Orthogonal Complements
I'm working on an exercise of Friedberg's linear algebra book. In the previous parts of the exercise, I proved that if $U$ is a unitary linear operator on an inner product space $V$, and $W$ is a ...
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Can there be a continuous function with infinite derivative everywhere?
Let $f:\mathbb{R}\to\mathbb{R}$ be continuous. For all nowhere-differentiable examples that I know of, for each $a\in\mathbb{R}$ there exist sequences $x_n\to a$ and $y_n\to a$ such that
$$\frac{f(...
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Function continuous nowhere whose domain and range are $[0,1]$
Find a function continuous nowhere, whose domain and range are both $[0,1]$.
My intuition was to start with $f(x)=x$ and exchange to $f(a)=b$, $f(b)=a$ for sufficiently many pairs of $(a,b)$. So I ...
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Counterexample request: $\sigma$-algebra with no smallest set containing a given (non-measurable) set?
To clarify, I am not asking for the definition of a $\sigma$-algebra generated by a certain (family of) subset(s).
Request: Given a set $X$, and a fixed subset $\tilde{X} \subseteq X$, a $\sigma$-...
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Can $\lim\limits_{(x,y)\to0}f(x)$ doesn't exist but at every path $y=\sum\limits_{k=1}^na_k x^k$, $\lim\limits_{(x,y)\to0}f(x)$ exist and are equal.
This is a generalization of this question
A quick and easy was to prove that a 2 dimensional limit like $$\lim\limits_{(x,y)\to0}\frac{xy}{x^2+y^2}$$ is to try 2 different linear paths and prove that ...
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Can $\lim\limits_{(x,y)\to0}f(x)$ doesn't exist but at every path $y=mx$, $\lim\limits_{(x,y)\to0}f(x)$ exist and are equal. [duplicate]
A quick and easy was to prove that a 2 dimensional limit like $$\lim\limits_{(x,y)\to0}\frac{xy}{x^2+y^2}$$ is to try 2 different linear paths and prove that they aren't equal or that the limit ...
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Series that is known to converge/diverge but for which all these standard tests are inconclusive .
I have noticed that nearly every series I have been asked to analyze its convergence or divergence can be handled by the usual collection of tests: the limit test, Cauchy condensation, the integral ...
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Example of ring with an element that is a left zero divisor, but not a right zero divisor
I understand that in a commutative ring (with unity) all left zero divisors are also right zero divisors. Do we just talk about zero divisors.
From Wikipedia I see that this is not the case if $R$ is ...
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Minkowski sum of the set of irrational with an open interval is open?
"If $A$ and $B$ are subsets of the set of real numbers where either $A$ or $B$ is open, then the Minkowski sum of $A$ and $B$ is open". I am failing to see how it can be true, as any real ...
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Can a function have a cusp at a point without being twice differentiable?
A function $f:[a,b]\to \mathbb{R}$ is said to have a cusp at a point $c$ if:
$f$ is continuous at $c$;
The one-sided derivatives satisfy
$$\lim_{x \to c^-} f'(x) = -\infty \quad \text{and} \quad \...
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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, ...
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Investigating the topological properties of a quotient space of $\mathbb R^2$, given by identification of irrational lines through $0$.
My question is about a very erratic quotient space. I encountered this space in some topology exercise. The space $X$ is described in the following:
Let $\mathbb R^2=\{(x,y):x,y\in \mathbb R\} $ be ...
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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 ...
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