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By the result of the previous paragraph, the limit function is also Lipschitz, with the same bound for the Lipschitz constant. In particular the set of all real-valued Lipschitz functions on a compact metric space X having Lipschitz constant ≤ K is a locally compact convex subset of the Banach space C(X).
Moreover, satisfies a Lipschitz condition on (,). If is bounded on (,), then it has both left-hand-side and right-hand-side derivative at every point in the interval (,). Moreover, the left-hand-side derivative is not greater than the right-hand-side derivative.
The particular case = is referred to as Lipschitz continuity. That is, a function is Lipschitz continuous if there is a constant K such that the inequality ((), ()) (,) holds for any ,. [15] The Lipschitz condition occurs, for example, in the Picard–Lindelöf theorem concerning the solutions of ordinary differential equations.
The number is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant (see proof below). If α = 1, then the function satisfies a Lipschitz condition. For any α > 0, the condition implies the function is uniformly continuous.
Both tests are the best of their kind. For the Dini-Lipschitz test, it is possible to construct a function f with its modulus of continuity satisfying the test with O instead of o, i.e. = (). and the Fourier series of f diverges. For the Dini test, the statement of precision is slightly longer: it says that for any function Ω such that
A set of functions with a common Lipschitz constant is (uniformly) equicontinuous. In particular, this is the case if the set consists of functions with derivatives bounded by the same constant. Uniform boundedness principle gives a sufficient condition for a set of continuous linear operators to be equicontinuous.
The limit function is also Lipschitz continuous with the same value K for the Lipschitz constant. A slight refinement is A slight refinement is A set F of functions f on [ a , b ] that is uniformly bounded and satisfies a Hölder condition of order α , 0 < α ≤ 1 , with a fixed constant M ,
The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied for k ≤ 1, then the mapping is said to be a non-expansive map. More generally, the idea of a contractive mapping can be defined for maps between metric spaces.