Does differentiability of a composite function imply differentiability of all its components? I.e. if $f(x)=g(x)+h(x)$ and we know $f(x)$ is differentiable at some point $x=a$, does this also imply $g(x)$ and $h(x)$ are differentiable there? Or is it possible that one of $g(x)$ and $h(x)$ (or both) are not differentiable at $x=a$, but in some way that is cancelled in the composite function, preserving the differentiability of $f(x)$ at $a$?
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How about $g(x) = |x|$ and $h(x) = -|x|$ – Tryss Jun 16 '15 at 05:41
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1Also any $f(x)$ and $-f(x)$. – Eli Rose Jun 16 '15 at 05:44
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Let $g = 1_\mathbb{Q}$ and $h = -g$. Then $f=g+h = 0$ is smooth, but $g,h$ are pretty far from smooth (with apologies to Marsellus).
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