In mathematical analysis, Lipschitz continuity is a regularity property of functions between metric spaces that is stronger than uniform continuity, and hence also stronger than continuity. 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; the smallest such bound is called the Lipschitz constant of the function (and is related to the modulus of uniform continuity). For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. We have the following chain of strict inclusions for functions over a closed and bounded interval of the real line with non-empty interior:
Continuously differentiable ⊂ Lipschitz continuous ⊂ α {\displaystyle \alpha } -Hölder continuous, where 0 < α ≤ 1 {\displaystyle 0<\alpha \leq 1} . We also have
Lipschitz continuous ⊂ absolutely continuous ⊂ uniformly continuous ⊂ continuous. Lipschitz continuity is named after German mathematician Rudolf Lipschitz.
Definitions Given two metric spaces (X, dX) and (Y, dY), where dX denotes the metric on the set X and dY is the metric on set Y, a function f : X → Y is called Lipschitz continuous if there exists a real constant K ≥ 0 such that, for all x1 and x2 in X,
d Y ( f ( x 1 ) , f ( x 2 ) ) ≤ K d X ( x 1 , x 2 ) . {\displaystyle d_{Y}(f(x_{1}),f(x_{2}))\leq Kd_{X}(x_{1},x_{2}).}
Any such K is referred to as a Lipschitz constant for the function f, and f may also be referred to as K-Lipschitz. The smallest constant is sometimes called the (best) Lipschitz constant of f or the dilation of f. The function f itself is sometimes referred to as a "Lipschitz map". If K = 1 the function is called a short map, and if 0 ≤ K < 1 and f maps a metric space to itself, the function is called a contraction. In particular, a real-valued function f : R → R is called Lipschitz continuous if there exists a positive real constant K such that, for all real x1 and x2,
| f ( x 1 ) − f ( x 2 ) | ≤ K | x 1 − x 2 | . {\displaystyle |f(x_{1})-f(x_{2})|\leq K|x_{1}-x_{2}|.}
In this case, Y is the set of real numbers R with the standard metric dY(y1, y2) = |y1 − y2|, and X is a subset of R. In general, the inequality is (trivially) satisfied if x1 = x2. Otherwise, one can equivalently define a function to be Lipschitz continuous if and only if there exists a constant K ≥ 0 such that, for all x1 ≠ x2,
d Y ( f ( x 1 ) , f ( x 2 ) ) d X ( x 1 , x 2 ) ≤ K . {\displaystyle {\frac {d_{Y}(f(x_{1}),f(x_{2}))}{d_{X}(x_{1},x_{2})}}\leq K.}
For real-valued functions of several real variables, this holds if and only if the absolute value of the slopes of all secant lines are bounded by K. The set of lines of slope K passing through a point on the graph of the function forms a circular cone, and a function is Lipschitz if and only if the graph of the function everywhere lies completely outside of this cone (see figure). A function is called locally Lipschitz continuous if for every x in X there exists a neighborhood U of x such that f restricted to U is Lipschitz continuous. Equivalently, if X is a locally compact metric space, then f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces that are not locally compact, this is a necessary but not a sufficient condition. More generally, a function f defined on X is said to be Hölder continuous or to satisfy a Hölder condition of order α > 0 on X if there exists a constant M ≥ 0 such that
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