Basic Analysis I — Real Analysis · Problem 6.1.11

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · uniform-convergence-averaging

Problem

Fix a continuous

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

. Let

f(x) \mathrel{:=} h(x)

for

x \in [a, b]

f(x) \mathrel{:=} h(a)

for

x < a

and

f(x) \mathrel{:=} h(b)

for all

x > b

. First show that

f : \mathbb{R} \to \mathbb{R}

is continuous. Now let

f_n

be the function

g

from Exercise 5.3.7 with

\epsilon = 1/n

, defined on the interval

[a, b]

. That is,

f_n(x) \mathrel{:=} \frac{n}{2} \int_{x-1/n}^{x+1/n} f.

Show that

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

converges uniformly to

h

[a, b]