Globally Lipschitz function

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 $|f(x)|\le |f(x_0)|+L|x-x_0|$, 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.