I'm trying to prove that the primitive function of an odd function is even without using integrals. It goes down to proving that $$g(x) = F(-x) - F(x) = 0$$
(Such that $F'(x) = f(x)$ and $f$ is odd)
I calculated $g'(x)$ but I keep getting $$g'(x) = -2F(x)$$ And the same goes for trying to prove that the primitive of an even function is odd.
Can you please point out where's my mistake ?