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][1] on "Existence/Uniqueness of solutions to quasi-Lipschitz ODEs". 


  [1]: https://mathoverflow.net/questions/55289/existence-uniqueness-of-solutions-to-quasi-lipschitz-odes