## 6.4. Topology and Continuity

### Theorem 6.4.6: Max-Min theorem for Continuous Functions

If

*f*is a continuous function on a compact set*, then***K***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 !