My lecture notes states the following lemma (sometimes called Cauchy's Theorem or Clairaut's Theorem) without proof:
Lemma. (Schwarz) Assume that $v_\xi$, $v_\eta$, and $v_{\xi\eta}$ exist and are continuous. Then $v_{\eta\xi}$ exists and $v_{\eta\xi}=v_{\xi\eta}$.
The context is the wave equation in one dimension $u_{tt} = c^2u_{xx}$, where we had considered the characteristic coordinates \begin{align}\xi &:= x+ct\\ \eta &:= x-ct \end{align} (without loss of generality, we take $c>0$), and we had written the solution in new coordinates to be $u(x,t) = v(\xi, \eta)$.
I know that we can prove the more general theorem, but what is a proof just for this easy lemma?