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In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants C ≥ 0, α {\displaystyle \alpha } > 0, such that | f ( x ) − f ( y ) | ≤ C ‖ x − y ‖ α {\displaystyle |f(x)-f(y)|\leq C\|x-y\|^{\alpha }} for all x and y in the domain of f. More generally, the condition can be formulated for functions between any two metric spaces. The number α {\displaystyle \alpha } 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. The condition is named after Otto Hölder. We have the following chain of inclusions for functions defined on a closed and bounded interval [a, b] of the real line with a < b : Continuously differentiable ⊂ Lipschitz continuous ⊂ α {\displaystyle \alpha } -Hölder continuous ⊂ uniformly continuous ⊂ continuous, where 0 < α ≤ 1. Hölder spaces Hölder spaces consisting of functions satisfying a Hölder condition are basic in areas of functional analysis relevant to solving partial differential equations, and in dynamical systems. The Hölder space Ck,α(Ω), where Ω is an open subset of some Euclidean space and k ≥ 0 an integer, consists of those functions on Ω having continuous derivatives up through order k and such that the kth partial derivatives are Hölder continuous with exponent α, where 0 < α ≤ 1. This is a locally convex topological vector space. If the Hölder coefficient | f | C 0 , α = sup x , y ∈ Ω , x ≠ y | f ( x ) − f ( y ) | ‖ x − y ‖ α , {\displaystyle |f|_{C^{0,\alpha }}=\sup _{x,y\in \Omega ,x\neq y}{\frac {|f(x)-f(y)|}{\|x-y\|^{\alpha }}},} is finite, then the function f is said to be (uniformly) Hölder continuous with exponent α in Ω. In this case, the Hölder coefficient serves as a seminorm. If the Hölder coefficient is merely bounded on compact subsets of Ω, then the function f is said to be locally Hölder continuous with exponent α in Ω. If the function f and its derivatives up to order k are bounded on the closure of Ω, then the Hölder space C k , α ( Ω ¯ ) {\displaystyle C^{k,\alpha }({\overline {\Omega }})} can be assigned the norm ‖ f ‖ C k , α = ‖ f ‖ C k + max | β | = k | D β f | C 0 , α {\displaystyle \|f\|_{C^{k,\alpha }}=\|f\|_{C^{k}}+\max _{|\beta |=k}\left|D^{\beta }f\right|_{C^{0,\alpha }}} where β ranges over multi-indices and ‖ f ‖ C k = max | β | ≤ k sup x ∈ Ω | D β f ( x ) | . {\displaystyle \|f\|_{C^{k}}=\max _{|\beta |\leq k}\sup _{x\in \Omega }\left|D^{\beta }f(x)\right|.} These seminorms and norms are often denoted simply | f | 0 , α {\displaystyle |f|_{0,\alpha }} and ‖ f ‖ k , α {\displaystyle \|f\|_{k,\alpha }} or also | f | 0 , α , Ω {\displaystyle |f|_{0,\alpha ,\Omega }\;} and ‖ f ‖ k , α , Ω {\displaystyle \|f\|_{k,\alpha ,\Omega }} in order to stress the dependence on the domain of f. If Ω is open and bounded, then C k , α ( Ω ¯ .... Discover the I B Holder popular books. Find the top 100 most popular I B Holder books.