Question

If f(x) is a function that satisfies the relation f(x+y)=f(x)+f(y) for all $x,yϵR,$ and if f'(0)=2 and function is differentiable every where, then find f(x) .

Answer 1

$f^{\prime }\left(x\right)={lim}_{h\to 0^{+}}\frac{f(x+h)-f\left(x\right)}{h}={lim}_{g{\to }^{+}}\frac{f\left(x\right)+f\left(h\right)-f\left(x\right)-f\left(0\right)}{h}(\because f(0)=0)$