This I found on homework 2: to determine the condition for the solution of

$$yu_{x}+xu_{y}=0$$

to be continuous at origin. Similar problems were covered in one previously lecture and several pictures were drawn. But even then I only vaguely felt those diagrams familiar (phase portrait covered in ODE), not their concrete connection. Please inform me how should I infer from the characteristic the condition for solution continuity, or even better, how to handle intersection of characteristics (as in another part of the question, where the circular characteristics shrink to a point).