If $f ◦f$ is differentiable, then $f ◦f ◦f$ is differentiable

Question

Let $ f : \mathbb R → \mathbb R$.

If f ◦ f is differentiable, then $f ◦ f ◦ f$ is differentiable?

how can I prove this? it feels like true by the chain rule .. but it feels also it is not always possible, there's have to be maybe a counterexample..

Answer 1

Take $f(x)=\frac 1 x, x \ne 0$ and $f(x) = 0, x = 0$ as a counterexample

Answer 2

Let $f(x) = \begin{cases} +x & x \in \mathbb{Q},\\ -x & x \not\in\mathbb{Q}\end{cases}$, then $f\circ f(x) = x$ is differentiable everywhere. However $f\circ f\circ f = f$ is not continuous and hence not differentiable for any non-zero $x$.