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We all know that if a function g is differentiable at a point a, and f is differentiable at g(a), then f∘g is differentiable at a with derivative f'(g(a))g'(a).

However, it is not necessary for both of them to be differentiable at their respective points. Consider f(x) = |x|, g(x)=0, a = 0. f is not differentiable at g(0)=0, but f∘g is differentiable since it is constant.

So, what are the conditions required for f∘g to be differentiable?

Aqeel
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  • The set of functions $f:\Bbb R\to\Bbb R$ such that $f\circ f=id$ is already impossibly large and complicated to describe. Take your favourite bijection $u:\Bbb R\setminus\text{Cantor set}\to\text{Cantor set}$ and call $f(x)=\begin{cases}u(x)&\text{if }x\notin \text{Cantor set}\ u^{-1}(x)&\text{if }x\in\text{Cantor set}\end{cases}$. – Sassatelli Giulio Dec 12 '23 at 09:58

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