Basic Analysis I — Real Analysis · Problem 3.4.14

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · uniform-continuity

Problem

Suppose for

f : [0, 1] \to \mathbb{R}

, we have

\bigl|f(x) - f(y)\bigr| \leq K\bigl|x - y\bigr|

for all

x, y

[0, 1]

, and

f(0) = f(1) = 0

. Prove that

\bigl|f(x)\bigr| \leq K/2

for all

x \in [0, 1]

. Further show by example that

K/2

is the best possible, that is, there exists such a continuous function for which

\bigl|f(x)\bigr| = K/2

for some

x \in [0, 1]