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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4
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58 views

If $f$ is continuous on $[a,b)$ and $[b,c]$, then $f$ is Riemann integrable on $[a,c]$.

True or False: If $f$ is continuous on $[a, b)$ and on $[b, c]$, then $f$ is Riemann integrable on $[a, c]$. I was unsure if the $)$ in $[a,b)$ completely changed the problem and made it false and I ...
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1answer
48 views

Show that the space of superharmonic functions is not a linear space

Let $(X_n)_{n\in\mathbb{N}_0}$ be a Markov chain with state space $E$ and transition matrix $P=(p_{i,j})_{i,j\in E}$. A real valued function $h$ on $E$ is called superharmonic if $h(x)\geq Ph(x)$ ...
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1answer
24 views

Lebesgue integrable discontinuity points

If a function is Lebesgue integrable, is it possible that it has as set of discontinuity points measure bigger than zero?
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3answers
112 views

A concave positive function on $[1,\infty)$ is uniformly continuous

Let $f$ be a concave positive function on $[1,\infty)$, then $f$ is uniformly continuous on $[1,\infty)$. This was a true or false problem that I couldn't prove to be true, so I'm thinking that maybe ...
1
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1answer
43 views

Arrange 10 points on five lines where each line(intersecting) has exactly 4 points

One possible case is that forming a star and then arranging 10 points on its vertices. Is there any other possible case for this arrangement? If not then how can we prove it mathematically? ...
4
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1answer
140 views

Generalized Riemann Integral: Bounded Nonexample?

Reference For a convergence theorem on integral see: Riemann Integral: Uniform Convergence For an improper version of integral see: Riemann Integral: Improper Version For a comparison of integrals ...
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1answer
46 views

Compactness of second-countability of $\omega$X$\omega_1$

Please discuss the following properties of the product space consisting of $\omega$X$\omega_1$: Is it compact? Is it 2nd countable? $\omega$ is the first infinite ordinal and $\omega_1$ is the ...
5
votes
1answer
52 views

An example of a function not in $L^2$ but such that $\int_{E} f dm\leq \sqrt{m(E)}$ for every set $E$

I am thinking about this problem: Let $f\in L^1 [0,1]$ to be a nonnegative function satisfied: $$\int_{E} f dm\leq \sqrt{m(E)}$$ for every measurable set $E\subset [0,1]$, Prove that $f\in ...
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3answers
62 views

Complex Mean Value Theorem: Counterexamples [closed]

This thread is just to collect some examples... Given an open domain $\Omega\subseteq\mathbb{C}$. Consider a holomorphic function $f:\Omega\to\mathbb{C}$. What would be a counterexample to: ...
0
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1answer
96 views

Are the following topological spaces locally compact?

I am trying to determine whether the following spaces are locally compact: a) the slotted plane b) the radial plane For part a) I am almost certain that it is not compact, but not sure how to go ...
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2answers
31 views

Non-unital commutative semigroups $S$ such that for all $x \in S$, the function $S \rightarrow S$ given by $y \mapsto x+y$ is a bijection?

Does there exist a commutative semigroup $S$ with the following (additively denoted) properties? For all $x \in S$, the function $S \rightarrow S$ given by $y \mapsto x+y$ is a bijection. $S$ has no ...
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144 views

The topology on $X/{\sim}\times X/{\sim}$ is not induced by $\pi\times\pi$.

I'm looking for an example of a topological space $X$ together with an equivalence relation $\sim$ where the product topology on $X/{\sim}\times X/{\sim}$ is not induced by $\pi\times\pi$ as a final ...
2
votes
1answer
24 views

$K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$.

Let $F$ be a field and $K$ be an extension field of $F$. Proof\Counterexample: $K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$. I haven't ...
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2answers
74 views

A topological space which is Frechet but not Strictly-Frechet.

Let $X$ be a topological space and $q \in X$. $X$ is strictly Frechet at $q$, if, for all $A_n \subset X, q \in \bigcap_{n \in \omega} \overline {A_n}$ implies the existence of a sequence $q_n \in ...
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1answer
22 views

Example of an increasing non-nonnegative sequence violating conclusion of monotone convergence theorem in space of finite measure

With Lebesgue measure in $\mathbb{R}$, $f_n(x) \equiv -\frac{1}{n}$ is a good example which doesn't coincide with MCT. However, I couldn't find another example when the measure is finite. Could ...
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1answer
83 views

If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$?

If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$? I'm asked to give a proof or a counterexample. I'm a bit confused on the notation of $\mathbb R(a)$, what does this ...
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1answer
36 views

Metrizable compact spaces and Hausdorff spaces with a countable network

I have two questions related to metrizable spaces and countable network ; Can we find a continuous mapping from a separable metric space onto a non metrizable compact Hausdorff space. If a Hausdorff ...
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0answers
34 views

Necessary and sufficient condition

could someone help me with 'necessary and sufficient' conditions problem with full proofs or counterexamples. thank you. It's either a 'necessary', 'sufficient', 'necessary and sufficient' or ...
0
votes
1answer
46 views

Cantor Intersection Theorem Without Closedness, counterexample

The Cantor Intersection Theorem is that Let $\{S_1,S_2,S_3,...\}$ be a countable collection of nonempty sets in $\mathbb R$ such that: $S_{k+1} \subset S_k$ for $k=1,2,3...$ Each $S_k$ ...
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2answers
120 views

Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that is differentiable only at $0$ and at $\frac{1}{n}$, $n \in \mathbb{N}$?

How to determine the existence of the function $f: \mathbb{R} \rightarrow \mathbb{R}$, which is differentiable only at $0$ and at $\frac{1}{n}$, $n \in \mathbb{N}$? It's more than enough to give an ...
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0answers
52 views

Bochner Integral vs. Lebesgue Integral

I'm trying to prove that for complex functions $f:\Omega\to\mathbb{C}$ that are not a priori measurable that: $$f\text{ Bochner integrable}\iff f\text{ Lebesgue integrable}$$ Basically it reduces to ...
4
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2answers
59 views

A category with arbitrary products, but not all limits, or finite limits not commuting with filtered colimits?

I'm interested in finding an example of a locally small category $\mathcal{C}$ having small filtered colimits and arbitrary small products but lacking, either all small limits, or either the ...
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2answers
59 views

Why does differentiability implies continuity, but continuity does not implies differentiability?

Why does differentiability implies continuity, but continuity does not implies differentiability? I am more interested in the part about a continuous function not being differentiable. Well, all ...
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0answers
23 views

Flow-Invariance w.r.t. $A$ doesn't invariance w.r.t. $\exp (tA)$

This answered question of mine explains that by saying consider $$\dot x =Ax,$$ then the stable and unstable subspaces are invariant with respect $A$ and therefore also with respect to $\exp (tA)$ one ...
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0answers
14 views

Iterated limits difficult example

Is there a fucntion $f:[-1,1]\subseteq \mathbb R^2 \to \mathbb R$ so that $$lim_{x\to 0}f(x,y)$$ exist $\forall y_0\in [-1,1]$ fixed and and $$lim_{y\to 0}f(x,y)$$ exist $\forall x_0\in [-1,1]$ ...
3
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2answers
27 views

Give an example of a function $f:D\subseteq \mathbb R^2 \to \mathbb R$ so that $lim_{x\to x_0}f(x,y)$ does not exist

Give an example of a function $f:D\subseteq \mathbb R^2 \to \mathbb R$ so that $$lim_{y\to y_0}f(x,y)$$ and $$lim_{x\to x_0}(lim_{y\rightarrow y_0}f(x,y))$$ exists but $lim_{x\to x_0}f(x,y)$ does not ...
14
votes
1answer
281 views

Is there a group which has precisely all finite groups as subgroups?

I would like to ask the following question: Does there exist a group $G$ such that every finite group can be embedded in $G$, and every proper subgroup of $G$ is finite? The closest ...
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2answers
36 views

Nonmeasurable Functions

Reference This question is related to: Banach Spaces: Uniform Integral vs. Riemann Integral Problem What are examples of real-valued functions: Bounded & Non-Step & Non-Measurable ...
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2answers
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Is a differentiable function on $(-2, 4)$ always integrable on $[-2, 4]$?

So my question is, say I have a function that is differentiable on $(-2, 4)$. Is it always integrable on $[-2, 4]$? I know that if $f$ is diff on $(-2, 4)$, then it is continuous on $(-2, 4)$. And I ...
3
votes
3answers
475 views

Continuous image of a locally connected space which is not locally connected

The question is pretty much in the title, I'm looking for an example of a locally connected space and continuous mapping such that the image is not locally connected. Thanks! EDIT: Corrected the ...
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1answer
58 views

Martingale $X_n \to \infty$ a.s.

Construct a martingale $X_n$ such that $X_n \to \infty $ a.s. I have trouble coming up with such an example and prove it. Can someone provide an example?
3
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1answer
61 views

Motivation and examples for ramification

I started learning algebraic number theory, but it seems like all the sources I had are too abstract, giving me difficulty understanding the concept and tripping me up frequently. For today it is ...
5
votes
2answers
362 views

Square root of compactly supported C-infinity function

Given $u \in \mathcal{C}^\infty_0(\mathbb{R}^n)$, $u \geq 0$ everywhere, is $v(x) = \sqrt{u(x)}$ also in $\mathcal{C}^\infty_0$? It is clear that the only problematic points are the boundary of the ...
5
votes
3answers
137 views

Totally disconnected topologies on countable set.

Are there totally disconnected topologies $\tau$ on a countable set $X$ such that $(X,\tau)$ is not homeomorphic to one of the following? $\mathbb{N}$ with the discrete topology; one-point ...
3
votes
2answers
186 views

Are quotient groups unique up to isomorphism

By this post, it seems quotient groups are unique up to isomorphism. is it correct? More clearly Let $G$ be a group and let $K,N\unlhd G$ be isomorphic normal subgroups. Are $\frac{G}{N}$ and ...
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6answers
355 views

Other guises for the vector space $\mathbb{R}^n$?

One way the vector space $\mathbb{R}^n$ can come up is as the space of polynomials over $\mathbb{R}$ of degree at most $(n-1)$ . Here we have the isomorphism: $$(a_0,a_1,\ldots,a_{n-1}) ...
2
votes
1answer
59 views

Counterexample for the infinitely many primes between two primes in a Noetherian ring

Consider the following Proposition: Proposition: Let $R$ be a noetherian ring. If $p_0 \subsetneq p_1 \subsetneq p_2$ is a chain of distinct prime ideals in $R$, then there exist infinitely many ...
0
votes
1answer
55 views

A set of real numbers whose limit points from a countable set

Construct a set of real numbers whose limit points from a countable set. Is the set you constructed closed? Is it compact? My example is $$G=\{1/n+1/m: n, m \in \mathbb N\}\cup \{0\}$$ and as ...
1
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1answer
40 views

Premeasures: Inner Measure vs. Outer Measure

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
51 views

Example of a Differential equation whose solution is not defined for all time $t$ [closed]

Give an example of a differential equation with its domain $R$ and an initial condition for this equation such the solution is not defined for all time $t$.
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1answer
30 views

Examples spectral sequence

I have to make a talk about spectral sequences, so I'd like to present some concrete examples of computation, after the general definition. I'd like to present three examples of spectral sequences: ...
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0answers
25 views

Function in Lipschitz space

I'm looking for a function that is in $W^{1,1}(0,1)$ but only in the Lipschitz space $\mathrm{Lip} (\alpha, L_2(0,1))$ for $0<\alpha < 1$. $\mathrm{Lip}(\alpha, L_2(0,1))$ is defined as the set ...
0
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1answer
58 views

Monoidal categories in which $\mathrm{Aut}(X \otimes Y) \cong \mathrm{Aut}(X) \sqcup \mathrm{Aut}(Y).$

I'm looking for examples of monoidal categories $\mathbf{C}$ such that one of the following two statements holds. For all objects $X$ and $Y$ of $\mathbf{C},$ $\mathrm{Aut}(X \otimes Y) \cong ...
2
votes
2answers
35 views

Dimension Field True/False.

I'm having trouble approaching how to determine truthfulness and falsehood of the following type of problems. $F$ and $K$ are fields. 1) Suppose that $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ ...
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2answers
55 views

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$ I'm pretty sure it's not necessarily true, but can't think of a counter example. Can you help me think of ...
2
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1answer
35 views

Example of topological space where there is a point and a subset $A $: $x \in \overline A $, but no sequence in $A $ converging to $x $?

It is known that if a space $X $ is metricable then for any subset $A $ of $X $ and a point $x \in \overline A $ there is a sequence of points in $A $ converging to $x $. I wonder if there is an ...
2
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1answer
72 views

Proof or counterexample: If $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ then $F(r)=F(r^3)$.

$F$ and $K$ are fields. Proof or counterexample: If $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ then $F(r)=F(r^3)$. I think I need to find a polynomial in $F(r^3)[x]$ that has $r$ as a root. I ...
9
votes
2answers
1k views

Give an example of a function $h$ that is discontinuous at every point of $[0,1]$, but with $|h|$ continuous on $[0,1]$

Give an example of a function $h:[0,1]\to\mathbb{R}$ that is discontinuous at every point of $[0,1]$, but such that the function $| h |$ that is continuous on $[0,1]$. I don't really even know where ...
0
votes
1answer
19 views

The set of differentiability of an extension from half-plane to the plane

This question is related to: Differentiability: Partially Defined Functions Consider a real-valued function $f:\mathbb{H}^2\to\mathbb{R}$. Are there some that admit no extension differentiable in ...
2
votes
2answers
51 views

Counterexample for “if every continuous function on $E$ is bounded, then $E$ is compact” [closed]

Give me counter example for this false statement: "Every continuous function on the set $E$ is bounded this implies $E$ is compact".