# lipschitz continuity

a function \(\mathbb{R}^n \to \mathbb{R}^m\) is locally lipschitz continuous at a point \(\hat{x}\) if

\begin{equation}
d(f(x_i), f(x_2)) \leq Ld(x_1, x_2) \forall x_1 x_2 \in B(\hat{x} \in \mathbb{R}^n, \delta) \exists \delta > 0 \exists L > 0
\end{equation}

Where \(B\) is the open ball, \(L\) is the lipschitz constant, and \(d\) is some measure of distance, typically the euclidean norm.

In a sense, this means that \(L\) is an upper bound on the change in distance between two points after being transformed by the function.

If the function is locally lipschitz continuous at all points \(\hat{x}
\in \mathbb{R}\) then we say it is *globally lipschitz continuous*.