Sets of local non-uniqueness points for autonomous scalar ODEs

It is known that if f : R → R is continuous then ẋ = f(x) with x(0) = x0 has at least one solution on some time interval [0, T (x0)). However, when f is not Lipschitz one cannot guarantee that the solutions are unique. In this paper we show that the set of points x0 ∈ R is a countable, nowhere dense set. (In particular, we show that if f(x0) 6= 0 then the solution must be locally unique.)