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 February 13th, 2017, 11:24 AM #1 Member   Joined: Jan 2016 From: Blackpool Posts: 40 Thanks: 0 Proving a continuous function is bounded How would i prove that the function F:[0,1]->Reals is bounded?
 February 13th, 2017, 02:40 PM #2 Senior Member   Joined: Sep 2016 From: USA Posts: 114 Thanks: 45 Math Focus: Dynamical systems, analytic function theory, numerics 1. Recall the definition of continuity. $F$ is continuous means that for any $\epsilon > 0$, for every $x$, there exists a $\delta$ (depending on $x$) such that $|F(y) - F(x)| < \epsilon$ whenever $y \in B_{\delta}(x)$. 2. Convince yourself of the following. Let $\delta(x)$ denote the appropriate $\delta$ for a specified $x$. Then $$\bigcup_{x \in [0,1]} B_{\delta(x)}(x)$$ is an open cover of $[0,1]$. 3. Recall that $[0,1]$ is compact. What does this say about the open cover above? 4. Profit. Thanks from Jaket1

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