Separately twice differentiable functions and the equation of string   oscillation
  We prove that for every separately twice differentiable solution $f$ of the PDE $f"_{xx}=f"_{yy}$ is of the form $f(x,y)=\phi(x+y)+\psi(x-y)$ for some twice differentiable functions $\phi, \psi$. 
Taras Banakh and Volodymyr Mykhaylyuk
math.AP