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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

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  • $\begingroup$ 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! $\endgroup$
    – Anil
    Commented Mar 31, 2017 at 14:53

1 Answer 1

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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.

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  • $\begingroup$ Thank you, I was going through literature in the field of Vector Calculus. It did not cross my mind to look at metric spaces. The definition presented on p. 154 (section 9.4, definition 9.4.1) of the book Metric Spaces (Searcóid, M. O., 2007) is in correspondence with your answer. $\endgroup$
    – Anil
    Commented Mar 31, 2017 at 14:51

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