Lebl-Basic-Analysis-I · Problem 3.4.14

Continuous Functions

Mathematics · uniform-continuity

Problem

Suppose for

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

, we have

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

for all

x, y

[0,1]

, and

f(0) = f(1) = 0

. Prove that

|f(x)| \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

|f(x)| = K/2

for some

x \in [0,1]