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The difference between uniform continuity and (ordinary) continuity is that, in uniform continuity there is a globally applicable (the size of a function domain interval over which function value differences are less than ) that depends on only , while in (ordinary) continuity there is a locally applicable that depends on both and . So uniform ...
Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant k is no longer necessarily less than 1). A contraction mapping has at most one fixed point.
Every Lipschitz continuous map is uniformly continuous, and hence continuous. 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 ...
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).
However, f is continuous if all functions are continuous and the sequence converges uniformly, by the uniform convergence theorem. This theorem can be used to show that the exponential functions , logarithms , square root function, and trigonometric functions are continuous.
A set A of continuous functions between two uniform spaces X and Y is uniformly equicontinuous if for every element W of the uniformity on Y, the set { (u,v) ∈ X × X: for all f ∈ A. (f(u),f(v)) ∈ W} is a member of the uniformity on X Introduction to uniform spaces
In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. [ 1 ]
A continuous function fails to be absolutely continuous if it fails to be uniformly continuous, which can happen if the domain of the function is not compact – examples are tan(x) over [0, π/2), x 2 over the entire real line, and sin(1/x) over (0, 1]. But a continuous function f can