Basic Analysis I — Real Analysis · Problem 6.2.18

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · lipschitz-uniform-convergence

Problem

(a)) Find a sequence of Lipschitz continuous functions on

[0, 1]

whose uniform limit is

\sqrt{x}

, which is a non-Lipschitz function. (b)) On the other hand, show that if

f_n : S \to \mathbb{R}

are Lipschitz with a uniform constant

K

(meaning all of them satisfy the definition with the same constant) and

\{f_n\}_{n=1}^\infty

converges pointwise to

f : S \to \mathbb{R}

, then the limit

f

is a Lipschitz continuous function with Lipschitz constant

K