I have $f \in C^0(\mathbb{R})$ local lipschitz, $(x_0, y_0) \in \mathbb{R}^2$ and $\lambda_{\text{max}}$ the maximal solution for the IVP,
$$ \begin{cases} y'(x) &= f(y(x)) \\ y(x_0) &= y_0 \end{cases} $$
defined on the maximal interval of existence $]I^{-}, I^{+}[$.
We assume $I^{+} < +\infty$.
I would like to show that,
$$ \lim_{x \rightarrow I^{+}} \lambda_{\text{max}}(x) = +\infty $$
where we have that,
$$ \lim_{x \rightarrow I^{-}} \lambda_{\text{max}}(x) = -\infty $$
Obviously, I'm not asking for the answer but a tip how to start. Because I don't really know where to start.
Roots of $f$ give constant solutions that can not be crossed.
Any non-constant solution is monotonous.
If a solution is bounded in space to one side, then it extends on that side to infinity in time on that side.
This should show that the claimed behavior is the only possibility left.