Examples on Picard-Lindelöf’s theorem for odes

Question

I am trying to find an example of ode, $x^{‘}=f(t,x)$, where $f$ does not satisfies Picard-Lindelöf theorem, but it still have unique solution.

Is it possible?

Answer 1

Yes, it is.

1. Osgood's criterion
2. $f(t,x)$ is decreasing as a function of $x$ for all $t$.