Let $f:\mathbb{R}\to \mathbb{R}$ be a $C^1$ function such that $f(n)=0$ for all $n \in \mathbb{Z}$. Solutions of $x'=f(x)$.

Question

I'm trying to do this exercise:

Let $f:\mathbb{R}\to \mathbb{R}$ be a $C^1$ function such that $f(n)=0$ for all $n \in \mathbb{Z}$.

Prove that the maximal solutions of $x'(t)=f(x(t))$ are defined and bounded for all $t \in\mathbb{R}$.

I really don't know how to attack this problem. I'd appreciate any hint.

Thanks for your time.

Answer 1

$y_n(t)=n$ are solutions. All solutions are unique, no two solutions can cross in a first order ODE.