If \( X \) is a compact space and \( f: X \to \mathbb{R} \) is continuous , then \( f \) is bounded and attains maximum and minimum values on \( \mathbb{R} \).
Proof
By the main theorem
, \( f(X) \) is compact and
due to Heine–Borel
, it is bounded and closed. In particular,
it contains its supremum and infimum.