Real Continuous Functions

Def Continuity
A function on is continuous at if for every such that , we have . Equivalently, it is continuous at if for all there exists such that if then . We say is continuous on if it is continuous at every .
e.g. Polynomial functions are continuous on .

Continuous Functions Theorems

Theorem

If is continuous, then is bounded, and both maximum and minimum exist.

Intermediate Value Theorem

If is continuous, then for any , there exists such that .

Proof Without loss of generality, assume . Let . As , is bounded. Let . By lemma, there exists a sequence such that . As for all , , we have . So . Now because