Given, $f(x),g(x),h(x)$ are three continuous and differentiable functions on $\Bbb{R}$ Which of the following statements are true if $$f(x+y)=g(x)+h(y)\qquad\forall(x,y)\in\Bbb{R}$$
a) $f'(0)=f'(1)$
b) $h(x)=g(x)$ if $h(0)=g(0)$
c) $g'(x)=h'(x)+1$
More than one option may be correct. I have no idea where to begin. Please help.
For part a), you should look at the definition of the derivative, as well as the relationship you are given:
$$\begin{align}f'(x) &= \lim_{\varepsilon \to0}\frac{f(x+\varepsilon) - f(x)}{\varepsilon} = \lim \frac{g(x) + h(\varepsilon) - g(x) - h(0)}{\varepsilon} \\&= \lim \frac{h(\varepsilon) - h(0)}{\varepsilon}\\ &= h'(0)\end{align}$$
So the derivative of $f$ at any point is equal to the derivative of $h$ at zero.
For part b) you should play around with $f(x) = f(x+0) = f(0+x)$. Inserting the definition of $f$ in terms of $g$ and $h$ should tell you how $g(x)$ and $h(x)$ are related.
For part c) you can start by noticing that b) and c) are mutually exclusive - if $h(0) = g(0) \implies h(x) = g(x)$, then $g'(x) = h'(x)$, and conversely that $g'(x) \neq h'(x) \implies g(x) \neq h(x)$.
