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February 13th, 2017, 12:24 PM  #1 
Senior Member Joined: Jan 2016 From: Blackpool Posts: 100 Thanks: 2  Proving a continuous function is bounded
How would i prove that the function F:[0,1]>Reals is bounded?

February 13th, 2017, 03:40 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 520 Thanks: 293 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. 

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bounded, continuous, function, proving 
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