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More generally, a set of functions with bounded Lipschitz constant forms an equicontinuous set. The Arzelà–Ascoli theorem implies that if {f n} is a uniformly bounded sequence of functions with bounded Lipschitz constant, then it has a convergent subsequence. By the result of the previous paragraph, the limit function is also Lipschitz, with ...
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.
Theorem (the Dini–Lipschitz test): Assume a function f satisfies = (). Then the Fourier series of f converges uniformly to f. In particular, any function that obeys a Hölder condition satisfies the Dini–Lipschitz test.
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 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.
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.
Lipschitz maps are particularly ... a quasimetric is defined as a function that satisfies all axioms for a ... i.e. a function satisfying the following conditions: ...
In mathematics, the Dini–Lipschitz criterion is a sufficient condition for the Fourier series of a periodic function to converge uniformly at all real numbers. It was introduced by Ulisse Dini ( 1872 ), as a strengthening of a weaker criterion introduced by Rudolf Lipschitz ( 1864 ).