TITLE: Uniqueness of IVP solution with a condition weaker than Lipschitz?
QUESTION [5 upvotes]: We know that Lipschitz condition with respect to $x$ in $$x' = f(t,x) , x(t_0)=x_0 $$ implies uniqueness of IVP problem above. Can we have uniqueness with condition less than Lipschitz?

REPLY [2 votes]: We have uniqueness in the separable case
$$f(x,t) = a(x) \cdot b(t)$$ 
if $a(x)$, $b(t)$ continuous and $a(x)\ne 0$ since for any solution $x(t)$ we can write 
$$\int_{x_0}^{x(t)} \frac{ dx}{a(x)}= \int_{t_0}^t b(t) dt $$