Basic Analysis I — Real Analysis · Problem 3.3.8

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · extreme-and-intermediate-value-theorems

Problem

Suppose

f : \mathbb{R} \to \mathbb{R}

is continuous and periodic with period

P > 0

. That is,

f(x + P) = f(x)

for all

x \in \mathbb{R}

. Show that

f

achieves an absolute minimum and an absolute maximum.