Lebl-Basic-Analysis-I · Problem 3.4.17

Continuous Functions

Mathematics · 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

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

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

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

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.