Question

If f+g is differentiable at a ,are f and g necessarily differentiable at a?If f.g and f are differentiable at a,what conditions on f imply that g is differentiable at a?

Answer 1

 if you allow $f$ and $g$ to be any continuous functions,

 then knowing that $f+g$ is differentiable will tell you nothing about the differentiability of $f$ and $g$ .

If you know that $f+g$ is differentiable and

 you assume that f is also differentiable while making no assumptions at all on g,

 then g will also be differentiable. (because $g=(f+g)−f$ and differences of diffentiable functions are differentiable.)

The most commonly referenced non-differentiable function is $g(x)=|x|$ . Let $f(x)=x.$ Then $f$ is differentiable, as is $fg$ .

Obviously, if $f$ and $fg$ are differentiable at a and $f(a)≠0$ then $g=\dfrac{f.g}{f}$ is differentiable at a by the usual quotient rule.