Lipschitz continuous

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The function LaTeX: f: X \rightarrow \mathbb{R} is Lipschitz continuous if there exists a value LaTeX: K, called the Lipschitz constant, such that LaTeX: |f(x) - f(y)| \le K \|x-y\| for all LaTeX: x and LaTeX: y in LaTeX: X. This relation is called the Lipschitz condition. It is stronger than continuity because it limits the slope to be within LaTeX: [-K, K]. The generalized Lipschitz condition is that there exists a monotonically increasing function, LaTeX: M : \mathbb{R} \rightarrow \mathbb{R} with the property that LaTeX: m(z) \rightarrow 0 as LaTeX: z \rightarrow 0 such that there exists LaTeX: K for which LaTeX: |f(x)-f(y)| \le K m(\|x-y\|) for all LaTeX: x and LaTeX: y in LaTeX: X.

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