A counter-example to differential function but not twice differential

Question

Find a function $f$ that is differentiable, but not twice differentiable and which does not belong to the following type: $$f(x) = \begin{cases} x^\alpha \sin(x^{\beta}) & x \neq 0 \\ 0 & x=0.\end{cases}$$ Please give me a hint.

Actually that class of functions is often used for examples of differentiable functions that are not $C^{1}$: do you mean that? — Dario, Jun 29 '14 at 11:26

score 4 · Answer 1 · answered Jun 29 '14 at 11:24

4

$x\mapsto x|x|$ should be a good candidate.

answered Jun 29 '14 at 11:24

Siminore

35,136

Gabriel Romon · Answer 2 · 2014-06-29T11:17:04.707

1

Consider $F$ an antiderivative of $x\to |x|$

Is $F$ twice differentiable?

edited Jun 29 '14 at 11:17

answered Jun 29 '14 at 11:11

Gabriel Romon

35,428
5
65
157

score 0 · Answer 3 · answered Jun 29 '14 at 11:40

0

Take the anti-derivative of the Weierstrass function which is continuous everywhere and differentiable nowhere.

answered Jun 29 '14 at 11:40

user1337

24,381

A counter-example to differential function but not twice differential

3 Answers3