Let $f:V\rightarrow \mathbb{R}^n$ be locally Lipschitz ($V$ is a subset of $\mathbb{R}\times\mathbb{R}^m\times \mathbb{R}^n$). Suppose we have a function $x:[t_0,\beta[\times W\rightarrow \mathbb{R}^n$ differentiable in the first argument ($W$ is an open subset of $\mathbb{R}$, $\beta$ is finite) such that for every $(t,\overrightarrow{\alpha})\in [t_0,\beta[\times W$ we have:

$$(t,\overrightarrow{\alpha},x(t,\overrightarrow{\alpha}))\in V$$

$$x_1(t,\overrightarrow{\alpha})=f(t,\overrightarrow{\alpha},x(t,\overrightarrow{\alpha}))$$

Here $x_1(t,\overrightarrow{\alpha})$ means partial derivative with respect to first argument.

It is also given that the function $g:W\rightarrow \mathbb{R}^n$ given by $g(\overrightarrow{\alpha})=x(t_0,\overrightarrow{\alpha})$ is locally Lipschitz.

**Question:** Does it follow that the function $x:[t_0,\beta[\times W\rightarrow\mathbb{R}^n$ is continuous ?

I can only prove the conclusion if the hypotheses are strengthened to $f,g$ Lipschitz instead of just merely locally Lipschitz.I would still like to know the answer in the locally Lipschitz case.

Thank you a lot.