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Lipschitz Continuity

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2020/06/20 Share



Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number.

Definition (Lipschitz Continuity). Given two metric spaces $(X, d_X)$ and $(Y, d_Y)$, where $d_X$ denotes the metric on the set $X$ and $d_Y$ is the metric on set $Y$, a function $f : X \mapsto Y$ is called Lipschitz continuous if there exists a real constant $K \geq 0$ such that, for all $x_1$ and $x_2$ in $X$,

If $K=1$, it’s called a short map.
If $0 \leq K < 1$ and $f$ maps a metric space to itself, the function is called a contraction.
In particular, a real-valued function $f : R \mapsto R$ is called Lipschitz continuous:

Definition (Lipschitz Continuous Gradient). A real-valued function $f$ is Lipschitz continuous gradient if:

Definition (Lipschitz Continuous Hessian). A real-valued function $f$ is Lipschitz continuous hessian if:

  1. 1. Definition