Question 9

Question

Why does a real valued continuous function obtain its maximum on a compact set?

Answer

Let’s say we have

• a continuous function $f:X\to \mathbb{R}$
• a compact set $X$

Proof/Reasoning

• The continuous image of a compact set is compact
  • Because inverse images of an open cover will again be an open cover.
• Then as a consequence of the Heine-Borel theorem will allow us to get a maximum on a compact set.
  • Because a continuous image of a compact set