calculus - Let $f$ be a function such that $f(ab)=f(a)+f(b)$ with $f(1)=0$ and derivative of $f$ at $1$ is $1$.

Let $f$ be a function such that $f(ab)=f(a)+f(b)$ with $f(1)=0$ and derivative of $f$ at $1$ is $1$

How can I show that $f$ is continuous on every positive number and

derivative of $f$ is $\frac{1}{x}$?