# Contradicting definitions of Lipschitz Continuity

Certain literature/lecture notes I've found have contradicting definitions of Lipschitz coninuity.

1. Let $F: \mathbb{R}^{m}\rightarrow\mathbb{R}^{m}$. Given an open set $B\subseteq\mathbb{R}^{m}$, $F$ is called Lipschitz continuous on the open set $B$ if $\exists L>0$ s.t. $\|F(\textbf{x}_1)-F(\textbf{x}_2)\|\leq L \|\textbf{x}_1-\textbf{x}_2\|, \forall \textbf{x}_1, \textbf{x}_2 \in B.$
2. Let $A\subset \mathcal{B}$ be a closed subset and $F$ a mapping $F: A\rightarrow A$. $F$ is called Lipschitz continuous on the closed set $A$ if $\exists L>0$ s.t. $\|F(\textbf{x}_1)-F(\textbf{x}_2)\|\leq L \|\textbf{x}_1-\textbf{x}_2\|, \forall \textbf{x}_1, \textbf{x}_2 \in A.$

I'm guessing whether the domain of $F$ is either closed or open does not matter, and also the domain and range of $F$ don't have to be of the same dimension. So, that would mean:

1. Let $V\subseteq \mathbb{R}^{n}$, and $F: V\rightarrow \mathbb{R}^{m}$. $F$ is called Lipschitz continuous on $V$ if $\exists L>0$ s.t. $\|F(\textbf{x}_1)-F(\textbf{x}_2)\|\leq L \|\textbf{x}_1-\textbf{x}_2\|, \forall \textbf{x}_1, \textbf{x}_2 \in V.$

Do you guys agree with definition number 3, and if so, what book backs this up?

Source:

1. "ADE (G1156) Spring 2006 Handout 3: Lipschitz condition and Lipschitz continuity", lecture notes of a Technical University of Eindhoven course, of which I don't know if is freely available for non-TU/e students/employees.

2. Gander, W., Gander, M. J. & Kwok, F. (2014). Scientific Computing: An Introduction using Maple and MATLAB (Vol. 11, Texts in Computational Science and Engineering). Springer International Publishing. - p.243

• Hey whoever made the remark about "$\forall \textbf{x}_{1}$, $\textbf{x}_{2} \in V$ should be $\forall \textbf{x}_{1}$, $\textbf{x}_{2} \in B$" in definition 1.: thank you, you were correct. Your comment got deleted after I edited my post, I did not know this would happen! – Anil Mar 31 '17 at 14:53

Lipschitz continuity can be defined between metric spaces. Let $(X_i,d_i)$ be two metric spaces. A function $f\colon X_1\to X_2$ is Lipschitz continuous if there is a constant $L>0$ such that $$d_2(f(x),f(y))\le L\,d_1(x,y)\quad\forall x,y\in X_1.$$ 1., 2. and 3. are particular cases of this general definition.