TITLE: ODE's without a Lipschitz condition
QUESTION [17 upvotes]: When teaching ODE's earlier this semester, one of my students asked the following question for which I didn't know the answer (and none of the textbooks I consulted seem to discuss it).  It is standard that if $f(x,y)$ is Lipschitz, then the ODE $y'=f(x,y)$ can be solved uniquely for any initial condition (at least locally).  The student asked for an example of a (non-Lipcshitz) $f(x,y)$ such that there is no solution.  Of course, one can give stupid examples (eg functions $f(x,y)$ that are so non-continuous that they cannot be the derivatives of anything).  However, I was unable to find a counterexample with $f(x,y)$ continuous.  Can someone provide one?

REPLY [5 votes]: I think it is worth mentioning that although it is true that Lipschitz  continuity guarantees unique solutions, there is a weaker regularity condition for $f$ which also gives uniqueness. The buzzword is "Osgood condition" and the theorem is given in Andrey Rekalo's answer to this MO question on "Existence/Uniqueness of solutions to quasi-Lipschitz ODEs".