Let f be a function defined on $R$ such that $f(xy)=f(x)+f(y)$; $x,y $ $\in R $

Question

Let $f$ be a function defined on $R$ such that $f(xy)=f(x)+f(y)$; $x,y $ $\in R $.

If f is differentiable at one point of $R$ then prove that f is differentiable on $R $.

Answer 1

Hint: Observe \begin{align} f(x\cdot 0)=f(0)+f(x)\ \ \Rightarrow \ \ f(x)=0. \end{align}