# Hölder condition

(redirected from Holder continuous)

## Hölder condition

[′hel·dər kən‚dish·ən]
(mathematics)
A function ƒ(x) satisfies the Hölder condition in a neighborhood of a point x0 if |ƒ(x) - ƒ(x0)| ≤ c |(x-x0)| n , where c and n are constants.
A function ƒ(x) satisfies a Hölder condition in an interval or in a region of the plane if |ƒ(x) - ƒ(y)| ≤ c|x - y |nfor all x and y in the interval or region, where c and n are constants.
References in periodicals archive ?
For example, Ren and Wu  discussed the convergence of the secant method under Holder continuous divided differences using a convergence ball.
The only difficulty is to show that the process [square root of V] is Holder continuous of order H.
The requirements of continuity and robustness stand for a whole different problem when disturbance is Holder continuous. The lack of integer-order differentiability of Holder functions suggests the design of fractional-order control since such functions exhibit well-posed derivatives just for some fractional orders.
In other words, almost every sample path of [??](t) is locally but uniformly Holder continuous with exponent v.
Assumptions like partially Lipschitz and linear growth, [alpha]-inverse Holder continuous or inverse Lipschitz, non-Lipschitz but bounded were used (see [16, 21, 22]).
The case when f is of bounded variation and u Holder continuous
By [C.sup.0,[alpha]](S) and [c.sup.0,[alpha].sub.T] (S) we denote the spaces of Holder continuous functions and Holder continuous tangential fields (0 < [alpha] < 1), respectively.
where k is a non-negative, bounded and Holder continuous function such that
The authors show that solutions of large classes of subelliptic equations with bounded measurable coefficients are Holder continuous. They present two types of results for such equations: results that generalize the Fefferman-Phong geometric characterization of subellipticity in the smooth case and results that generalize a case of Hormander's algebraic characterization of subellipticity for sums of squares of real analytic vector fields.
Corollary 4.8 Let X have a bounded density [f.sub.X], which is Holder continuous with exponent [alpha] [member of] (0,1].
Furthermore, we have that d[[micro].sub.w] (x) = f (x) dx, where f [member of] [C.sup.[infinity]] (U.sup.L.sub.l=1] ([a.sub.l],[b.sub.l])) and f is Holder continuous on [S.sub.w] with exponent 1/2.
The technique we use here is well-known as the Bishop 1/4-3/4 method, but we include the details because we will sharpen them in section 5 to obtain Holder continuous peak functions.
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