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**zebra2147** Let $\displaystyle f:[0,1]\rightarrow \mathbb{R}$ be a function that maps closed and bounded intervals onto closed and bounded intervals, hence if $\displaystyle 0\leq a<b\leq 1$, then for some real numbers $\displaystyle c\leq d, f([a,b])=[c,d]$.

1)Prove that there is an $\displaystyle x_{0}$ contained in $\displaystyle [0,1] $such that for all $\displaystyle x$ contained in $\displaystyle [0,1], f(x)\leq f(x_{0}).$

2)Prove the range of $\displaystyle f$ has the intermediate value propery.

3)Give an example showing $\displaystyle f$ not needing to be continuous.

My attempts/observations...

1) It would appear that we are looking for an upper bound for $\displaystyle f$? I'm not reall too sure how to start that one though...

2) Well, possibly we can define $\displaystyle f(0)$ and $\displaystyle f(1)$ and then use that for $\displaystyle 0\leq a<b\leq 1$, $\displaystyle c\leq d, f([a,b])=[c,d]$. Then show that $\displaystyle [c,d]$ is between $\displaystyle f(0) and f(1)$???

3)I think I can probably take a better guess once I have a better understand of 1) and 2)

Any help would be greatly appreciated.