Basic Analysis I — Real Analysis · Problem 3.4.17

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · uniform-continuity

Problem

Suppose

f : [a, b] \to \mathbb{R}

is a function such that for every

c \in [a, b]

there is a

K_c > 0

and an

\epsilon_c > 0

for which

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

for all

x

and

y

(c - \epsilon_c, c + \epsilon_c) \cap [a, b]

. In other words,

f

is ``locally Lipschitz.'' a) Prove that there exists a single

K > 0

such that

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

for all

x, y

[a, b]

. b) Find a counterexample to the above if the interval is open, that is, find an

f : (a, b) \to \mathbb{R}

that is locally Lipschitz, but not Lipschitz.