Separately twice differentiable functions and the equation of string oscillation

We prove that for every separately twice differentiable solution $f$ of the PDE $f{\mathrm{"}}_{xx}=f{\mathrm{"}}_{yy}$ is of the form $f(x,y)=\varphi (x+y)+\psi (x-y)$ for some twice differentiable functions $\varphi ,\psi$.