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Is there example of two real functions $f$ and $g$ such that: $g$ has a local minimum at $x=0$ ($g$ is not necessarily differentiable at $x=0$), $f\circ g$ is differentiable at $x=0$ but $(f\circ g)'(0)\neq 0$, and $f$ is differentiable at $g(0)$?

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    $g$ can be even discontinuos and the local min point $x=0$ can be an end-point of the domain.2012-10-02

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Sure -- let $f(x)=x(x+1)$ and $ g(x) = \begin{cases} -1 & x=0 \\ x & x \ne 0 \end{cases} $

It gets more difficult if you want $g$ to be continuous.

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    I am writing a paper about the derivative notion.I wanted to invalidate the chain rule under certain assumptions. Thank you again!2012-10-02