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I am not sure how to proceed with this question:

Construct counterexamples for the following statements.

(a) If a function $g(x)$ is differentiable at $x=a$ and a function $f(x)$ is not differentiable at $g(a)$, then the function $(f\circ g)(x)$ is not differentiable at $x=a$.

(b)If a function $g(x)$ is not differentiable at $x=a$ and a function $f(x)$ is differentiable at $g(a)$, then the function $(f\circ g)(x)$ is not differentiable at $x=a$.

(c) If a function $g(x)$ is not differentiable at $x=a$ and a function $f(x)$ is not differentiable at $g(a)$, then the function $(f\circ g)(x)$ is not differentiable at $x=a$.

For (a) I have begun by outlining what I know:

  • g'(a) exists

  • f'(g(a)) does not exist

  • (f\circ g)'(x)=f\;'(g(x))\cdot g'(x)

Which leaves me stuck because then (f\circ g)'(a)=f\;'(g(a))\cdot g'(a) and it is stated that $f(x)$ is not differentiable at $g(a)$.

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    Thanks. I failed to think ahead when recommending we stay continuous.2011-10-27

1 Answers 1

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Functions which aren't differentiable due to discontinuity are bound to pose problems, so try something continuous like $f(x)=|x|$. Not differentiable at $x=0$, but what if $g$ is a constant function, say $g(x)=0$? Try thinking along that sort of line.