The Extreme Value Theorem

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We examine a fact about continuous functions.

(a): A function $f$ has an global maximum at $x=a$ , if $f(a)\ge f(x)$ for every $x$ in the domain of the function.
(b): A function $f$ has an global minimum at $x=a$ , if $f(a)\le f(x)$ for every $x$ in the domain of the function.

A global extremum is either a global maximum or a global minimum.

If we are working on an finite closed interval, then we have the following theorem.

Extreme Value Theorem If $f$ is a continuous function for all $x$ in the closed interval $[a,b]$ , then there are points $c$ and $d$ in $[a,b]$ , such that $(c,f(c))$ is a global maximum and $(d,f(d))$ is a global minimum on $[a, b]$ .

Below, we see a geometric interpretation of this theorem.

Would this theorem hold if we were working on an open interval?

yes no

Consider $\tan (\theta )$ for $-\pi /2 < \theta < \pi /2$ . Does this function achieve its maximum and minimum?