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February 13th, 2017, 11:24 AM   #1
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Proving a continuous function is bounded

How would i prove that the function F:[0,1]->Reals is bounded?
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February 13th, 2017, 02:40 PM   #2
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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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