Let $f:\mathbb{R} \rightarrow \mathbb{R}$, $f \in C^{1}$,
and $f(0)=f(1)=0$, $f(x)>0$ for $x \in (0,1)$.
Prove that solution (with maximal domain) $u$ of problem:
$x'=f(x)$,
$x(0)=x_{0} \in (0,1)$
satisfy
$\lim_{t \rightarrow -\infty}$ $u(t)=0$
$\lim_{t \rightarrow +\infty}$ $u(t)=1$
Thanks a lot for your help.