# Tagged Questions

Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) ...

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### Condition on subsets of normed linear space such that “every real valued continuous function on the subset is uniformly continuous” imply boundedness

If $A$ is a connected subset of a real normed linear space such that every real valued continuous function on $A$ is uniformly continuous , then is it true that $A$ is bounded ? If not , then what if ...
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### What is the definition and relationship between the following continuity? Also, does such definition exist?

continuity of a finite sequence continuity of a infinite sequence continuity of a finite series continuity of a infinite series continuity of a function continuity of a linear operator continuity of ...
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### Can a function $f:\Bbb R\to \Bbb R$ be discontinuous at all minus one point? [duplicate]

Can a function $f:\Bbb R\to \Bbb R$ be discontinuous at all minus one point? I know functions can be discontinuous everywhere, continuous everywhere but nowhere differentiable, etc. But I'm not sure ...
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### Solving the boundary value problem?

Let $G(x,y)$ be the Green's function of the boundary value problem $$[(1+x)u']'+(\sin{x})u=0,~x\in[0,1],~u(0)=u(1)=0.$$ Then, the function $g$ defined by $$g(x)=G(x,\dfrac{1}{2}),~x\in[0,1].$$ is ...
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### Give an example in which $a_0=a_1<a_2=a_3<a_4=a_5<a_6=\cdots$ for the bisection method. [on hold]

Give an example in which $a_0=a_1<a_2=a_3<a_4=a_5<a_6=\cdots$ for the bisection method. I don't see how to make an example of this form. I'm sure there is something simple/elegant that does ...
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### Alternative proof for dominated convergence theorem without using Fatou's lemma?

The conclusion of dominated convergence theorem is that $||f-f_n||_{L^1}\to 0$ as $n\to\infty$. After showing that $f_n\to f\in L^1$, why is it not possible to use the continuity of norm in order to ...
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### Continuity of the Fourier transform of a measure

If $\mu$ is a complex finite Borel measure on a separable real Hilbert space $H$ then $$x \mapsto \hat \mu (x) = \int \limits _H \Bbb e ^{\Bbb i \langle x, y \rangle } \Bbb d \mu _{(y)}$$ is ...
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### Laplace Transform of uniformly convergent series

Let $\sum_{n=1}^{\infty} f_n(x)$ be a uniformly convergent series of functions each of which has a laplace transform defined for $s \geq \alpha$. Show that $f(x)=\sum_{n=1}^{\infty} f_n(x)$ has a ...
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### Show that a function is continuous on an infinite interval

We may show that a function is continuous over an interval $[a,b]$ by applying $$\lim_{x \to a+} f(x) = f(a) \quad \text{and} \quad \lim_{x \to b−}f(x) = f(b)$$ But what about on an interval ...
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### Is there any non-compact space $X$ such that every maximal ideal of $C(X, \mathbb R)$ is of the form $\{f \in C(X, \mathbb R) : f(a)=0 \}$?

Does there exist a non-compact metric space $X$ , such that for every maximal ideal $M$ of $\mathcal C(X, \mathbb R)$ , $\exists a \in X$ such that $M:=\{f \in \mathcal C(X, \mathbb R) : f(a)=0 \}$ ...
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### Prove that the continuous image of closed and bounded interval is a closed and bounded interval. [closed]

Let $f$ be a nonconstant function that is continuous on $[a,b]$, where $a < b$. Prove that the image of $f$ is some closed bounded interval $[c,d]$, where $c < d$.
### If $\sup f(S) = f(\sup S)$ and $\inf f(S) = f(\inf S)$, $f$ is continous
Suppose $f: \mathbb{R} \to \mathbb{R}$ is a function. Suppose for any subset $S \subset \mathbb{R}$, $$\sup f(S) = f(\sup S)$$ and $$\inf f(S) = f(\inf S)$$ Prove that $f$ is continous. Attempt: ...