Show that if $f$ is a twice differentiable, continuously differentiable real-valued function on an open interval in $E^2$ and $\partial f ^2 / \partial x \partial y= 0$ then there are continuously differentiable real-valued functions $f_1, f_2$ on open intervals in $\Bbb R$ such that
$$f(x,y) = f_1(x) + f_2(y)$$
I have an idea of how to prove this but I cannot seem to formulate a rigorous proof. Essentially what I want to say is that $\partial f / \partial y$ is equal to something in terms of only $y$, meaning the $x$ and $y$ terms of $f(x,y)$ are separate.