Question

Give an example to show that Rolle’s theorem can fail if f is merely assumed to be almost everywhere differentiable, even if one adds the additional hypothesis that f is continuous. This example illustrates that everywhere differentiability is a significantly stronger property than almost everywhere differentiability. We will see further evidence of this fact later in these notes; there are many theorems that assert in their conclusion that a function is almost everywhere differentiable, but few that manage to conclude everywhere differentiability.

Name: Problem 3, Chapter 1: An Introduction To Measure Theory (January 2011 Draft)
Uploaded: 2022-03-31T15:21:05-08:00
Duration: 1 min 34 s
Channel: Lucas Finney

   Give an example to show that Rolle’s theorem can
fail if f is merely assumed to be almost everywhere differentiable,
even if one adds the additional hypothesis that f is continuous. This
example illustrates that everywhere differentiability is a significantly
stronger property than almost everywhere differentiability. We will
see further evidence of this fact later in these notes; there are many
theorems that assert in their conclusion that a function is almost everywhere differentiable, but few that manage to conclude everywhere
differentiability.

An Introduction To Measure Theory (January 2011 Draft)

Terence Tao 1st Edition

Chapter 1, Problem 3

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Give an example to show that Rolle’s theorem can fail if f is merely assumed to be almost everywhere differentiable, even if one adds the additional hypothesis that f is continuous. This example illustrates that everywhere differentiability is a significantly stronger property than almost everywhere differentiability. We will see further evidence of this fact later in these notes; there are many theorems that assert in their conclusion that a function is almost everywhere differentiable, but few that manage to conclude everywhere differentiability.