I don't know either about what happens to uniqueness and it's becomming a very interesting problem for me: If g(t,y,y') and its partials are continuous at a point then we can be assured of a unique solution about the point. However, the proof of such I am familiar with makes no comment about uniqueness if they are not continuous.
I don't know what to do if f is not 1. Working it numerically "close to zero" is unacceptable to you right?
Maybe/hopefully someone else here more knowledgeable than I will post a comment and we can both learn.