Let $I\subseteq \mathbb{R}^{n}$ be an arbitrary (not necessarily closed) interval and $f:I\times \mathbb{R}^{n}\to \mathbb{R}^{n}$ a continuous function such that in $I\times \mathbb{R}^{n}$ satisfies a global Lipschitz condition on its second variable. Then is it true that:

For every point $(a,b)\in I\times \mathbb{R}^{n}$ there exists a a solution to the equation $y^{\prime}=f(x,y)$ defined over the entire $I$

Picard-LindelĂ¶f theorem states that there exists a solution in a local closed neighborhood $[a-\epsilon,a+\epsilon]$. One could try to glue the local solutions to get a global one but then there will be a problem with the boundary of the resulting (possibly) open interval.

Does such a globally defined solution always exist? Are there references in the literature on this?

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