Extreme_Value_Theorem

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Definition

If $f:[a,b] \rightarrow \mathbb{R}$ is continuous on the closed interval $[a,b]$, then $f$ attains a maximum and minimum value. That is, there exist $x_0, x_1 \in [a,b]$ such that $f(x_0) \leq f(x) \leq f(x_1)$ for all $x \in [a,b]$.

Proof

Since $[a,b]$ is compact and $f$ is continuous, the image $f([a,b])$ is also compact, hence closed and bounded.

Let $M = \sup f([a,b])$. Since $f([a,b])$ is closed and $M$ is a limit point of $f([a,b])$, we have $M \in f([a,b])$.

Thus there exists $x_1 \in [a,b]$ with $f(x_1) = M$. A similar argument works for the minimum.