page_title: Continuous function
section_title: Uniform, Hölder and Lipschitz continuity
hierarchical_section_title: Continuous function / Continuous functions between metric spaces / Uniform, Hölder and Lipschitz continuity
caption_attribution_description: English: A Lipschitz continuous function. There is a double cone (shown in white) whose vertex can be translated along the graph of the function, so that the graph will always be entirely outside the cone.
context_page_description: In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, sufficiently small changes in the input of a continuous function result in arbitrarily small changes in its output. If not continuous, a function is said to be discontinuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.
  Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. A stronger form of continuity is uniform continuity. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.
context_section_description: The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way δ depends on ε and c in the definition above. Intuitively, a function f as above is uniformly continuous if the δ does
  not depend on the point c. More precisely, it is required that for every real number ε > 0 there exists δ > 0 such that for every c, b ∈ X with dX(b, c) < δ, we have that dY(f(b), f(c)) < ε. Thus, any uniformly continuous function is continuous. The converse does not hold in general, but holds when the domain space X is compact. Uniformly continuous maps can be defined in the more general situation of uniform spaces.
  A function is Hölder continuous with exponent α (a real number) if there is a constant K such that for all b and c in X, the inequality
  holds. Any Hölder continuous function is uniformly continuous. The particular case α = 1 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 b, c in X. The Lipschitz condition occurs, for example, in the Picard–Lindelöf theorem concerning the solutions of ordinary differential equations.
caption: For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone.
exif: {}