Is the following derivative application statement true or false?

Question

Determine if the following statement is true or false. Provide proof if true, or a counterexample if false.

Statement:

If $(f\circ g)(x)$ is differentiable, then $f(x)$ and $g(x)$ must be differentiable.

I think that this statement is false and my counterexample is as follows: let $f(x) = |x|$, $g(x) = x^2$, however, I wanted to find a different example that doesn't make use of $x$, $x^2$, or any constant functions, but I can't seem to find one.

Could anybody give me a hand, please?

Answer 1

Take any bijection $f:\mathbb R\to\mathbb R$, and consider its inverse. Then $f\circ f^{-1}$ is the identity function which is obviously differentiable, but $f$ need not be.

Answer 2

Let $\text{sign}(x) = 1$ if $x >= 0$, and $-1$ if $x<0$

Let $f(x) = -\text{sign}(x)$

Let $g(x) = 1$ if $x<0$ else $g(x)=x$

Then $(f \circ g)(x) = 1$

Answer 3

$$ f(x) =g(x)= \begin{cases} 1, & \text{if $x$ is rational} \\ 0, & \text{if $x$ is irrational} \end{cases}$$

I wanted to mention this even though $f\circ g$ is constant because it seems cool to me that neither $f$ nor $g$ are continuous anywhere.