Basic Analysis I — Real Analysis · Problem 7.5.12

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · continuous-functions-metric-spaces

Problem

Let

C\bigl([a,b],\mathbb{R}\bigr)

be the set of continuous functions and

C^1\bigl([a,b],\mathbb{R}\bigr)

the set of once continuously differentiable functions on

[a,b]

. Define

d_C(f,g) \coloneqq \|f-g\|_{[a,b]} \quad \text{and} \quad d_{C^1}(f,g) \coloneqq \|f-g\|_{[a,b]} + \|f'-g'\|_{[a,b]},

where

\|\cdot\|_{[a,b]}

is the uniform norm. By Example 7.1.8 and Exercise 7.1.12, we know that

C\bigl([a,b],\mathbb{R}\bigr)

with

d_C

is a metric space and so is

C^1\bigl([a,b],\mathbb{R}\bigr)

with

d_{C^1}

. a) Prove that the derivative operator

D: C^1\bigl([a,b],\mathbb{R}\bigr) \to C\bigl([a,b],\mathbb{R}\bigr)

defined by

D(f) \coloneqq f'

is continuous. b) On the other hand if we consider the metric

d_C

C^1\bigl([a,b],\mathbb{R}\bigr)

, then prove the derivative operator is no longer continuous. Hint: Consider

\sin(nx)