# Existence of solutions for $x'=f(t,x)$ for $f$ not necessarily $C^1$, but with other conditions

Let $$f:\mathbb{R}\times\mathbb{R}^n \to \mathbb{R}^n$$ be a continuous, locally Lipschitz function that satisfies the following condition:

$$|f(t,x)|\leq C(1+|x|)\; , \forall (t,x)\in \mathbb{R}\times\mathbb{R}^n$$

Show that the IVP $$x' = f(t,x),\; x(t_0)=x_0$$ admits a unique solution.

We can get existence of solutions from Peano's existence theorem. It's also easy to see that $$C(1+|x|)$$ is globally Lipschitz. Though, I don't know how to connect the dots. It feels like we would need $$f$$ to be globally Lipschitz to be able to say anything about uniqueness of solutions...

• The case where $f$ doesn't depend on $t$ is proved as Proposition IV.3 in Zehnder's book Lectures on Dynamical Systems, using Grönwall's lemma. Probably something similar works in the general case. Oct 10 '19 at 19:32

What is usually associated with a linear bound of $$f$$ of the given form is that the domain of any solution is $$\Bbb R$$. But your task description does not even mention that?