Let $f,g:\Bbb R\to\Bbb R$ be real variable continuous functions and define $h:=f+g$; suppose $h$ non-constant.
If $h$ is differentiable at some $x_0\in\Bbb R$, does the same holds for both $f$ and $g$? Or there exist a couple of functions $f,g$ not differentiable at $x_0$ such that their sum does?
Maybe is trivial but I can't find counterexamples.
In what case is this true?
$\endgroup$ 42 Answers
$\begingroup$As others have pointed out, 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.)
$\endgroup$ $\begingroup$If you know what a group and a subgroup are, think that the set of differentiable functions is a proper subgroup of the group of the continuous functions.
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