Let $f : \mathbb{R} \to \mathbb{R}$ an uniformly continuous function on $\mathbb{R}$. We consider the Cauchy's problem : $y'(t) = f(y(t))$ and $y(0)=y_0$.
Then every maximal solution is defined on $\mathbb{R}$.
First thing is that we can consider a maximal solution $y : I \to \mathbb{R}$ according to Cauchy-Peno-Arzela's theorem (because $f$ is an an uniformly continuous function on $\mathbb{R}$ so continuous on $\mathbb{R}$).
Moreover, as $f$ is an uniformly continuous function on $\mathbb{R}$ it enables us to find $a, b\in \mathbb{R}$ such that $\vert f(x) \vert \le a\vert x\vert + b$ for all $x\in \mathbb{R}$.
With that inequality we can deduce that every maximal solution of the Cauchy's problem is global (in our case $y$ will be defined on $\mathbb{R}$). Indeed, it is a consequence of the "blow up in finite time" property (I think I can apply it with the hypothesis of uniform continuity) and of the Grönwall's lemma.
Does it hold ?
Thanks in advance