Extreme Value Theorem

Theorem

If $f$ is Continuous on a Closed Interval Then, there is an Global Extrema on that interval

Alternate Theorem

Suppose $C\subset {\mathbb{R}}^n$ is closed and bounded. If $f:C\subset {\mathbb{R}}^n\to \mathbb{R}$ is Continuous, then there are points $x_0,x_1\in C$ where $f$ attains its Global Maxima and Global Minima.

Open Interval No EVT

An open interval does not necessarily have a max/min. Consider the example where the absolute max/min would be at the endpoints. There would always be a max/min bigger than the existing max/min.

Proof

• Extreme Value Theorem Proof