# Tagged Questions

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### $f'$ is bounded and isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\lim_{x\to y}f'$ does not exist

Prove/disprove: $f$ has a bounded derivative and $f'$ isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\displaystyle\lim_{x\to y}f'$ does not exist. I think that if $f'$ ...
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### Rudin: A compact metric space $K$ has a countable base, therefore $K$ is separable.

Hi this is a problem from Rudin's Princ. of Mathematics. I was hoping someone could check this part of my proof for the following question, comments would be very appreciated!: $25.$ Prove that ...
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### Using Properties of a Dense Set to prove characteristics of a continuous function

1. If $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous and $f(x)=0$ for all $x$ in a dense set $E$, then $f(x)=0$ for all $x \in \mathbb{R}$ 2. If $f:\mathbb{R} \rightarrow \mathbb{R}$ and ...
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### For what points $c$ in $\mathbb{R}$ is $f$ continuous?

Let $X \subset \mathbb{R}$ be a fintie set and define $f:\mathbb{R} \rightarrow \mathbb{R}$ by $f(x)=1$ is $x \in X$ and $f(x)=0$ otherwise. At which points $c \in \mathbb{R}$ is $f$ continuous? ...
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### Proof of uniform continuity on compact sets

Show that a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that is continuous on a compact set $K$ is uniformly continuous on $K$. Is the proof below correct? Proof: Let $\epsilon > 0$ and let ...
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### If F is real-entire, then how to write, $F(z)- F(w)$ in terms of $(z-w)$ and $(\bar{z}- \bar{w})$?

Define $F:\mathbb C \to \mathbb C$ such that $F(z)= \sum_{j,k=0}^{\infty}c_{j,k} z^{j} \bar{z}^{k}$ is an entire real analytic function on $\mathbb C$ with $F(0)=0.$ My question is :How to show: ...
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### Compact operator on invariant subspace is compact

Statement: Let $T \in \mathscr{B}(\mathscr{H})$, where $T$ is a compact operator. Let $M$ be a closed invariant subspace of $T$. Show that the restriction of $T$ to $M$ is compact. Attempted Proof: ...
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### Using Cauchy's integral formula to find the best estimate for $|f^{(n)}(0)|$ under a condition

Question: Is the following proof valid? Ahlfors: If $f(z)$ is analytic for $|z| < 1$ and $|f(z)| \le 1/(1-|z|)$, find the best estimate of $|f^{(n)}(0)|$ that Cauchy's inequality will yield. ...
I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ ...