Let $f$ be a function ​satisfying such that $f(x+y)=f(x)+f(y)$ Let $f$ be a function ​satisfying such that $f(x+y)=f(x)+f(y)$ and $f(x)=x^2g(x)$ for all $x$ and $y$, $g(x)$ is continuous function.then find $f''(x)$.
As we can see that $f(ax)=af(x)$, so can we take $f(x)=ax$? But if do so, then there is no use of $g(x)$. What am I missing here?

If you have already proved that $f(ax)=af(x)$ for all $x$, then the fact that $f(x+y)=f(x)+g(x)$ tells you that $f(x)$ is linear. All linear functions have constant first derivative (as you can check from the definition of derivative), and hence vanishing second derivatives, i.e. $f''(x)=0$ for all $x$.
