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 $g(0)$?
Composition of non-differentiable functions
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real-analysis
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0$g$ would have to be non-differentiable at $0$. Otherwise $(f\circ g)^{\prime}(0) = f^{\prime}(g(0))\cdot g^{\prime}(0) = 0$ since $g$ has a local minimum at $0$. – 2012-10-02
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0$g$ can be even discontinuos and the local min point $x=0$ can be an end-point of the domain. – 2012-10-02