Solving $u_{\xi\eta} = 0$ and differentiability conditions on solutions

Question

After transformation someone often encounters the PDE $$ u_{\xi\eta} = 0 $$ but I am quite confused about the differentiability conditions of its solution (for example in this post I read different versions). I know it could be solved by first integrating with respect to $\eta$, yielding $$ u_{\xi} = g(\xi) $$ for some differentiable function $g$, then integration with respect to $\xi$ yields $$ u = G(\xi) + f(\eta) $$ for some function $f$, and everything that is needed is that $G(\xi)$ is differentiable, and for $g$ and $f$ I do not see that any differentiability conditions are necessary. Also if I solve $$ u_{\eta\xi} = 0 $$ which is (by Schwartz) equivalent to the above I get everything reversed, namely $$ u = g(\xi) + F(\eta) $$ where $F$ has to be differentiable, but in the above mentioned post it is said in the answer that the general solution is $$ u = F(\xi) + G(\eta) $$ where $F,G \in C^2$, in other sources, for example this wiki entry on d'Alembert formula, I read that it has the general solution $$ u = F(\xi) + G(\eta) $$ for $F,G \in C^1$. So everywhere I read it differently... So could anyone explain to me how these differentiability conditions follow, what are the correct conditions, and how they relate to the above mentioned derivations?