Continuous functions on compact sets are bounded

A continuous function $f:K\to\mathbb R$ on a compact topological space $K$ is bounded: there exists $M\geq0$ such that

|f(x)|\le M\qquad(x\in K).

Proof

If $K$ is empty the assertion is vacuous. Otherwise the extreme value theorem gives a minimum $m$ and maximum $M_+$ ; take $M=\max\{|m|,|M_+|\}$ .

Scope

No metric on the domain is required. Applied to a compact subset of another space, continuity means continuity for the subspace topology.