# Tagged Questions

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### Does $f_n(a_n)\to f(a)$ hold?

Say, we have $f, f_n \in C^0(\mathbb R, \mathbb C)$ such that $f_n \xrightarrow{\text{uniform}}f$ and a sequence of reals $a_n \to a$. Does it then hold that $f_n(a_n)\to f(a)$? I couldn't think of ...
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### Continuity of f [on hold]

Is the statement below true.If it is could someone provide a proof of this.If its not provide a counter example $f(x)$ is continuous at $x_0$ $\implies \exists \delta>0:$ (if ...
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### Question about limit points in relation with continuity and functional limits

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I have the feeling that the author is being careless about limit points in his theorems or I am not understanding ...
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### A continuous mapping $f:\mathbb{R}\rightarrow\mathbb{R}$ may have a fixed point?

Let a function $f:\mathbb{R}\rightarrow\mathbb{R}，$satisfied $$\forall x,y\in\mathbb{R},|f(x)-f(y)|\leq k|x-y|.(0<k<1)$$ Prove: There exists a only one $\xi\in \mathbb{R}$ ,such that ...
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### Characterization of continuity in terms of preimages of open sets

1--8 Theorem. If $A\subset \mathbb R^n$, a function $f:A\to \mathbb R^m$ is continuous if and only if for every open set $U\subset \mathbb R^m$ there is some open set $V\subset \mathbb R^n$ such ...
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### If a function is bounded and the variable is bounded, is the function continuous?

Suppose you have a function $f:C\to \mathbb R$ where $C$ is closed and bounded interval and $f$ is bounded. Does that mean $f$ is continuous? I know the other way around (if $f$ is continuous, $f$ ...
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### Are there standard parameters for the Weierstrass nowhere differentiable function?

On Wikipedia, the Weierstass non-differentiable function is defined as: $$f(x)=\sum^{\infty}_{n=0}a^n\cos(b^n\pi x)$$ where $0<a<1$, $0<b$, and $ab>1+\frac 32 \pi$ Since it seems like, ...
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### In the semi linear uniform space

In the semi linear uniform space, If $f$ is a function from $(X ,Γ_X)$ to ($Y,Γ_Y)$ where $f(x_n)$ converges to $f(x)$ whenever $x_n$ converges to $x$,show that $f$ is continuous at $x$.
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### Problem related to Mean Value Theorem

I found out a question that I can't figure out a way to solve it. Plz can anyone help me. Question is, Prove that $\exists\,C\in(0,\pi/4)\,\mathrm{s.t.}\,\tan(\pi/4+C)=3/C$ I know this should be ...
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### Requirement for continuity of unit normal vector

When considering a subset $\Omega \subset \mathbb{R}^{n}$. If we consider $\nu$, the outward unit surface normal to $\partial \Omega$, what are the requirements of $\partial \Omega$ which will ...
I need help in this question. I figured out a way to solve the question but not sure the proof is valid. This is the question, Given $a \in\mathbb{R}$, and a function ...