Basic Analysis I — Real Analysis · Problem 6.2.2

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · uniform-convergence-derivative-limit

Problem

Let

f_n(x) \mathrel{:=} \dfrac{x^n}{n}

. Show that

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

converges uniformly to a differentiable function

f

[0, 1]

(find

f

). However, show that

f'(1) \neq \lim_{n \to \infty} f_n'(1)