= Globally Lipschitz function
= Globally Lipschitz
{synonym}
A <function> is globally Lipschitz when one finite <Lipschitz bound> holds for every pair in its entire specified <domain of a function>. The <sine> and the <absolute value> have this property on the <real line>. For any reference point $x_0$, a globally <Lipschitz continuous> real function obeys $|f(x)|\leq|f(x_0)|+L|x-x_0|$, so its size grows at most linearly in the distance from $x_0$.
The <function> $f(x)=x^2$ is a <locally Lipschitz function> on the <real line>, but is not globally Lipschitz there: pairs $x=n+1$, $y=n$ have ratio $2n+1$, which is unbounded. Restricting the domain to $[-R,R]$ does give a global bound $2R$. Global and local describe where one bound must work, rather than two intrinsic labels independent of the domain.
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