What is derivative a of a continuous function such that $f(x+y) = f(x) +f(y)$? Let $f: \mathbb{R} \to \mathbb{R} $ be a continuous function with $f(x+y) = f(x) +f(y),\forall x,y\in \mathbb{R}.$ 
Find $\frac {df} {dx} $, if it is exist?

This is called Cauchy's functional equation. You can show that $f(0)=0$, and if you let $c=f(1)$ then you can deduce that $$f(2)=f(1+1)=f(1)+f(1)=2c\qquad \text{ and } \qquad f\left(\frac{1}{2}\right)=\frac{f(1)}{2}= \frac{c}{2} $$
This extends similarly to
$$f(x)=cx$$ for all rational $x$, but because $f$ is continuous, $f(x)=cx$ for all real $x$. You should be able to show that the derivative of this linear function exists.
