Basic Analysis I — Real Analysis · Problem 6.2.1

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · uniform-limit-not-differentiable

Problem

Find an explicit example of a sequence of differentiable functions on

[-1, 1]

that converge uniformly to a function

f

such that

f

is not differentiable. Hint: There are many possibilities, simplest is perhaps to combine

|x|

and

\frac{n}{2}x^2 + \frac{1}{2n}

, another is to consider

\sqrt{x^2 + (1/n)^2}

. Show that these functions are differentiable, converge uniformly, and then show that the limit is not differentiable.