# Tagged Questions

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### Question in the Continuity of a function

I have this function: $$(J''(u)v,w)=(v,w)-(KN_{f'}(Ku)Kv,w)$$ for all $u,v,x\in L^2[0,1]$ such that $f\in C^1([0,1]\times\mathbb{R},\mathbb{R})$ and $Ku(t)=\int_0^1 G(t,s) u(s)ds$ $K$ is symetric, ...
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### Applications of Vito Volterra's theorem

We know from Volterra's theorem that: There cannot exist two pointwise discontinuous functions on an interval $(a,b)$ for which the continuity points of one, are the discontinuity points of the ...
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### Continuity and differentiability of $x^a\sin ({1\over x})$ at $0$

Consider the function $$g_a (x) = \begin{cases} x^a\sin ({1\over x}) & x \neq 0 \\ 0 & x=0 \end{cases}$$ I am looking to determine for which $a$ the map $g_a$ is differentiable on ...
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### If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,…)$, prove that $f(x)=0$ on $[0,1]$.

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,...)$, prove that $f(x)=0$ on $[0,1]$. This is what I have, how does it look? Proof: Let $P(x)$ be any ...
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### $\{f_n\}$ equicontinuous sequence of functions on compact $K$, converges pointwise on $K$ then converges uniformly on $K$.

Suppose $\{f_n\}$ is an equicontinuous sequence of functions on a compact set $K$ and $\{f_n\}_{n=1}^{\infty}$ converges pointwise on $K$. Prove that $\{f_n\}_{n=1}^{\infty}$ converges uniformly on ...
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### Compact Domain and Inverse Image

I am trying to show that given $f:M \rightarrow N$, where $M$ is compact, $f$ is continuous and onto, then given $A \subset N$: $$f^{-1}(A) \text{ closed} \implies A\text{ closed}$$ I am dealing ...
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### Monotone convergence of continuous functions

Is it true that for a sequence of functions $f_n \in C(K,\mathbb{R})$, where $K \subset \mathbb{R}^n$ the limit is semicontinuous? I would say that if it's a monotone decreasing lsequence, then the ...
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### Compactness implies Continuity?

I am stuck on this question (probably there are many counterexamples, but I can't find any). "Suppose $f:\mathbb{R}\mapsto\mathbb{R}$ that preserves compactness (i.e, for every $K \subseteq R$, then ...
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### Finding examples of continuous functions

I'm looking for 1) a function that is discontinuous at 0, 1, 1/2, 1/3, 1/4, 1/5, ... but continuous everywhere else 2) a function that is discontinuous at 1, 1/2, 1/3, /4, 1/5, ... but continuous ...
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### Proving discontinuity

Assume set $A$ is countable and let$$f(x)=\cases{1 \text{ if }x\in A\\0\text{ if }x\notin A }.$$ Prove that $f$ is not continuous at $c\in A$. I've seen such a problem before where ...
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### The rational numbers can't be the set of points on which a real function is continous

Let $A$ be the set of all points on which $f:\mathbb{R}\rightarrow \mathbb{R}$ is continous. How can I prove that $A$ can never be $\mathbb{Q}$? edit: also, is it possible to generalize it for any ...
I understand that in an open interval the only functions that are continuous but not uniformly are functions whose limits are singularities. But when we have a function $f:H\rightarrow\mathbb{R}$ and ...