Extreme Value Theorem

analysis

Definition

Extreme Value Theorem

Let be a continuous function on a closed and bounded interval . Then attains both an absolute maximum and an absolute minimum at least once.

That is, there exist points such that for all :

Generalisation

The theorem can be generalised to higher dimensions: a continuous function from a compact set to the real numbers is bounded and attains its extrema. In , by the Heine-Borel theorem, a set is compact if and only if it is closed and bounded.

Necessity of Conditions

• Continuity: If is not continuous, it may not attain extrema (e.g., a function with a jump discontinuity at its peak).
• Closed Interval: On an open interval , a function like does not attain a maximum or minimum.
• Boundedness: On an unbounded set, a function like does not attain a maximum.