Continuity of Poincare Maps

Question

Consider the system $\dot{x}=f(x,t)$, where $f$ is continuously differentiable in both $x$ and $t$.
And let $f(x,t)=f(x,t+T)$.
The Poincare Map $P$ maps the value of $x(t=0)$ to $x(t=T)$, i.e, $$P(x_0)=x_T$$ Is $P(x_0)$ continuous in $x_0$? If not, what are the conditions sufficient for it to be continuous?
I wanted an answer to this question, and the answer given there, as far as I can tell, assumes the Poincare Map is continuous.
Thanks in advance!