For this question (eqn 14) I found the characteristic curve to be $\frac{x}{y} = C$. It asks what the difference is between the cases $(x,y)\neq (0,0)$ and that the solution be continuous at (0,0). From what I understand, in the former case we can simply say the solution is any funtion of one variable, $u(x,y) = \phi (\frac{x}{y})$, on the domain $(x,y)\neq (0,0)$. In the second case, we need to pick some value for $u(0,0)$ to make it continuous, and elsewhere it's the same as the other case. Is this a sufficient explanation? It feels lacking to me. Could someone offer assistance?