If $g$is differentiable at $a$ and $g(a)=0$ and $h$ is continuous at $a,$ then $f=g.h$ is differentiable at $a$

Question

If $g$ is differentiable at $a$ and $g(a)=0$ and $h$ is continuous at $a,$ then $f=g.h$ is differentiable at $a,$ whether $h$ is differentiable there or not.

My working: I can prove this statement as long as LHD and RHD of $h$ are finite at $a$. How to prove it for general case.

score 1 · Accepted Answer · answered Sep 27 '19 at 03:27

1

Note that $f(a)=0$. Now, given any $x\neq 0$,

$$\frac{f(a+x)-f(a)}{x}=\frac{g(a+x)h(a+x)}{x}=h(a+x)\frac{g(a+x)-g(a)}{x}$$

What happens if $x\to 0$?

answered Sep 27 '19 at 03:27

Reveillark

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Ohh thanks a lot – Makar Sep 27 '19 at 03:39

If $g$is differentiable at $a$ and $g(a)=0$ and $h$ is continuous at $a,$ then $f=g.h$ is differentiable at $a$

1 Answers1