# Tagged Questions

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### A continuous function that attains neither its minimum nor its maximum at any open interval is monotone

Let $f: \mathbb R\to \mathbb R$ be a continuous function such that $f$ attains neither its minimum nor its maximum at any open interval $I \subseteq \mathbb R$ , then how to prove that $f$ is ...
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### $C^0$ is a closed subspace of $L^{\infty}$

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. Let $f\in C^0(\bar\Omega)$. I have to prove that $\|f\|_{\infty}=\|f\|_{L^{\infty}}$. One implication is trivial. Let's consider the other one. ...
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### If $f$ is continuous, so is $g=|f|$ [closed]

Prove that if $f$ is continuous, so is $g=|f|$. I need help on this. Thank you. Ok, this is my first time here. The definition of continuity i am using is that $f$ is continuous at $a$ if for any ...
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### how to show that $f_n$ is nonnegative on an open interval for all $n$ large enough

Let $\{f_n\}_{n=1}^\infty$ be a sequence of continous functions on $[0,1]$ and for all $x\in [0,1], f_n(x)$ is eventually nonnegative. Show that there is an open interval $I\subseteq[0,1]$ such that ...
Suppose f(x) is continuous and bounded on (0,1). Is f(x) uniformly continuous on (0,1)? I think yes, because it's bounded, i.e. there exists $M: |f(x)| < M$. We could use this M as $\delta$ in the ...