Provide an example of a function $f: \mathbb{R} \to \mathbb{R}$ which is not differentiable but $f^{2}$ is differentiable.

Question

A question on a practice exam asks:

This confuses me because $f$ is differentiable when $f^2$ is differentiable seems to prove that any differentiable $f^2$ means that f would be differentiable.

$f(x)=|x|$ is not differentiable at $x=0$, but $f(x)^2=x^2$ is differentiable at $x=0$. — conditionalMethod, Dec 08 '19 at 02:34

score 3 · Answer 1 · answered Dec 08 '19 at 02:34

3

The function $f:x\rightarrow |x|$ is a possible answer.

answered Dec 08 '19 at 02:34

Arthur Breton

1

Note that the page you've shared says that is $f^2$ is differentiable at a point $\textit{and}$ doesn't vanish at this point then $f$ is also differentiable at that point. In the case of $f:x\rightarrow |x|$, the function is not differentiable at 0 and vanishes at the same point. – Arthur Breton Dec 08 '19 at 02:37

score 1 · Answer 2 · answered Dec 08 '19 at 02:33

1

Hint: let $f(x)=1$ when $x$ is rational and $f(x)=-1$ when $x$ is irrational.

answered Dec 08 '19 at 02:33

Botond

score 1 · Answer 3 · answered Dec 08 '19 at 02:33

1

Continuity is needed as a premise.

Consider $f(0)=-1$ and $f(x)=1$ for $x\ne 0$ about its differentiability at zero.

answered Dec 08 '19 at 02:33

user284331

3 Answers3