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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.

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closed as off-topic by Loïc Teyssier, Mikhail Katz, Pedro Lauridsen Ribeiro, Pace Nielsen, coudy Feb 11 '18 at 20:47

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  • $\begingroup$ Why not use the "local" property to prove the existence of local solutions and then glue them together using uniqueness on the overlaps of a suitable cover. $\endgroup$ – user64472 Feb 7 '18 at 13:14
  • $\begingroup$ Crossposted to MSE: math.stackexchange.com/questions/2458822/… $\endgroup$ – Dap Feb 9 '18 at 12:50