Separately Twice Differentiable Functions and the Equation of String Oscillation

Abstract

We prove that for every separately twice differentiable function \(f:\mathbb {R}^2\to\mathbb{R}\) with that \(f''_{xx}=f''_{yy}\) there exist twice differentiable functions \(\varphi, \psi:\mathbb R\to\mathbb{R}\) such that \(f(x,y)=\varphi(x+y) + \psi(x-y)\).