Lipschitz continuity

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In mathematics, more specifically in real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions which is stronger than regular continuity. Intuitively, a Lipschitz continuous function is limited in how fast it can change; a line joining any two points on the graph of this function will never have a slope steeper than a certain number called the Lipschitz constant of the function.

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.

The concept of Lipschitz continuity can be defined on metric spaces and thus also on normed vector spaces. A generalisation of Lipschitz continuity is called Hölder continuity.



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 (for example, Y might be the set of real numbers R with the metric dY(x, y) = |xy|, and X might be a subset of R), a function

is called Lipschitz continuous if there exists a real constant K ≥ 0 such that, for all x1 and x2 in X,

Any such K is referred to as a Lipschitz constant for the function ƒ. The smallest constant is sometimes called the (best) Lipschitz constant; however in most cases the latter notion is less relevant. If K = 1 the function is called a short map, and if 0 < K < 1 the function is called a contraction.

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 x1x2,

For real-valued functions of several real variables, this holds if and only if 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).

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