Basic Analysis I — Real Analysis · Problem 4.2.5

The Derivative — Jiri Lebl

Mathematics — Real Analysis · lipschitz-squared-constant-function

Problem

Suppose

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

is a function such that

\bigl|f(x) - f(y)\bigr| \leq \bigl|x - y\bigr|^2

for all

x

and

y

. Show that

f(x) = C

for some constant

C

. Hint: Show that

f

is differentiable at all points and compute the derivative.