Basic Analysis I — Real Analysis · Problem 3.3.3

Continuous Functions — Jiri Lebl

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

Problem

Let

f : (0, 1) \to \mathbb{R}

be a continuous function such that

\lim_{x \to 0} f(x) = \lim_{x \to 1} f(x) = 0

. Show that

f

achieves either an absolute minimum or an absolute maximum on

(0, 1)

(but perhaps not both).