Let $\Omega \subset \mathbb{R} \times \mathbb{R}^n$ be an open (non-empty) set. Let $f \; : \; \Omega \to \mathbb{R}^n$ be a continuos function such that
$$\forall (t,x) \in \Omega \;\;\exists \delta > 0 \; : \; ]t-\delta,t+\delta[ \times B_{\delta}(x) \subset \Omega \text{ and } \forall t' \in ]t-\delta,t+\delta[,\forall x_1,x_2 \in B_{\delta}(x) \;\; |f(t',x_1) - f(t',x_2)| \leq L|x_1-x_2|$$
So basically $f$ has to satisfy the hypothesis of Cauchy-Lipschitz Theorem (the hypothesis in the link are a bit stronger then the one I'm using, I'm assuming the weaker hypothesis that $f$ is just continuos and locally Lipschitz respect to $x$ uniformly respect to $t$ )
Let also $u \; : \; ]\alpha,\beta[ \to \mathbb{R}^n$ ( with $\alpha < \beta$) be a $C^1$ function such that
$\forall t \in ]\alpha,\beta[ \;\; (t,u(t)) \in \Omega \text{ and } u'(t) = f(t,u(t))$
let's also assume that $\beta < +\infty$ (I don't care about $\alpha$ ) and that $u$ can't be extendend over $\beta$ and still satisfy the condition stated above, is it possible that both the conditions (A) and (B) aren't true ?
$$(A) \lim_{t \to \beta^-}{|u(t)|} = +\infty$$ $$(B) \lim_{t \to \beta^-}{d((t,u(t)),\partial \Omega)} = 0$$
where
$$d((t,u(t)),\partial \Omega) = \inf\bigg\{ \sqrt{ (t-s)^2 + |u(t) - x|^2 } \; : \; (s,x) \in \partial \Omega \bigg\}$$
(Clearly by both I mean that both the conditions are not true at the same time)
If the answer is no, prove it, if it's yes, provide an example.
Observe that, if the answer is no, then verifying that either (A) or (B) isn't verified is enough to prove that $u$ can be extended over $\beta$, which could be useful when studying the maximal interval of existence of a solution of an IVP.