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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. The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way δ {\displaystyle \delta } depends on ε ...
This shows uniformly continuous functions are not always differentiable. Despite being nowhere differentiable, the Weierstrass function is uniformly continuous. Every member of a uniformly equicontinuous set of functions is uniformly continuous.
A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).
For a Lipschitz continuous function, there exists a double cone (white) whose origin can be moved along the graph so that the whole graph always stays outside the double cone. In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions.
A sublinear modulus of continuity can easily be found for any uniformly continuous function which is a bounded perturbation of a Lipschitz function: if f is a uniformly continuous function with modulus of continuity ω, and g is a k Lipschitz function with uniform distance r from f, then f admits the sublinear module of continuity min{ω(t), 2r ...
The sum and difference of two absolutely continuous functions are also absolutely continuous. If the two functions are defined on a bounded closed interval, then their product is also absolutely continuous. [4] If an absolutely continuous function is defined on a bounded closed interval and is nowhere zero then its reciprocal is absolutely ...
A fundamental result in the theory of approximately continuous functions is derived from Lusin's theorem, which states that every measurable function is approximately continuous at almost every point of its domain. [4] The concept of approximate continuity can be extended beyond measurable functions to arbitrary functions between metric spaces.
The restriction of any continuous function to any subset of its domain (dense or otherwise) is always continuous, so the conclusion of the Blumberg theorem is only interesting for functions that are not continuous. Given a function that is not continuous, it is typically not surprising to discover that its restriction to some subset is once ...