Let $h:\mathbb{R}^3 \to \mathbb{R} $.
Consider the following integro-differential equation: $$g'(t)=\int_0^t h(t,g(t),g(z))\;dz $$
With initial condition $g(0)=0$, is a unique solution for $g:[0,\infty) \to \mathbb{R}$ implied by the continuity of $h$?
Thoughts: Seems like the needed condition is for the RHS to be Lipschitz. That way, I think the usual results should carry over and a unique solution should exist. Not exactly sure how the integral affects the problem.
Edit 1: I think it is probably necessary to assume that $h$ is Lipschitz or at least Lipschitz in the second argument to assure a Lipschitz condition on the RHS. Is this sufficient for uniqueness?