By Picard's theorem, it is known the IVP $$ y'=f(x,y),\quad y(x_0)=y_0 $$ has a unique solution (which is also stable) in some interval $[x_0-h,x_0+h]$ for some $h>0$, provided $f$ is continuous in $x$ and uniformly Lipchitz continuous in $y$ (by uniform we mean the Lipchitz constant is independent of $x,y$).
I am very curious to know if the above condition is also sufficient, that is if the above IVP has a unique stable solution, will the function $f$ be continuous in $x$ and uniformly Lipchitz continuous in $y$?
Any help would be appreciated.
Thanks.