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The question goes like this:

If $f(x)$ is a non-constant, continuous function defined on a closed interval $[a,b]$ Then by the Extreme Value Theorem, there exist an absolute minimum $m$ and an absolute maximum $M$.

Based on this, I need to show that the range of $f$, $\{f(x) \mid a \le x \le b\}$, is the interval $[m, M]$.

Thanks in advance!

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closed as off-topic by Eevee Trainer, Thomas Shelby, egreg, José Carlos Santos, Leucippus Apr 1 at 1:02

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Consider the Intermediate value theorem, let $x\in [m,M]$ and then since there are $x_1,x_2$ such that $f(x_1)=m, f(x_2)=M$ we get the desired result.

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First show that the range is contained in the interval $[m,M]$. Then use the Intermediate Value Theorem to show that if $y \in [m,M]$ then there exists an $x \in [a,b]$ with $y=f(x)$.

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