Derivative of a continuous function. Let $g:R\to R$ be a continuous function with $g(x+y)=g(x)+g(y), \forall x,y\in R.$ Find $\frac{dg}{dx},$ if it exist.

First we have $g(0) = 2g(0)\to g(0)=0$. Then it follows that
$$g'(x) = \lim_{y\to 0}\frac{g(x+y)-g(x)}{y} = \lim_{y\to 0}\frac{g(y)}{y} = \lim_{y\to 0}\frac{g(y)-g(0)}{y} = g'(0)$$
As noted below in the comments, this only shows that if $g'(x)$ exist for one $x$ then it exist for all $x$ and is indeed a constant.
