Is there any difference in a function being locally Lipschitz on $\mathbb{R^n}$ and being globally Lipschitz?
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$\begingroup$A function $f$ is locally Lipschitz if for each compact subset $K \subset \mathbf{R}^n$, there is a constant $M = M_K$ such that $$|f(x) - f(y)| \le M|x-y|$$ for all $x, y \in K$. The Lipschitz constant $M$ can in general depend on the subset $K$; in particular, it could get worse as we take $K$ larger and larger. Consider for example the function $f(x) = x^2$ on $\mathbf{R}$.
We say $f$ is globally Lipschitz if there is a constant $M$ such that $$|f(x) - f(y)| \le M|x-y|$$ for all $x, y \in \mathbf{R}^n$. Equivalently, $f$ is globally Lipschitz if $f$ is locally Lipschitz but it has a Lipschitz constant $M$ that works for all compact sets $K$.
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