Theorem: Max/Min theorem for Continuous Functions

• If f is a continuous function on a compact set K, then f has an absolute maximum and an absolute minimum on K.
• In particular, f must be bounded on the compact set K.

Proof:

With the work we have done previously, this proof is easy: Since K is compact and f a continuous function, f(K) is compact also. The compact set f(K) is bounded, so that f is bounded on K. The compact set f(K) also contains its infimum and supremum, so that f has an absolute minimum and maximum on K.

That's all !