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The definition of uniform continuity appears earlier in the work of Bolzano where he also proved that continuous functions on an open interval do not need to be uniformly continuous. In addition he also states that a continuous function on a closed interval is uniformly continuous, but he does not give a complete proof.
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).
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 ...
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.
An isomorphism between uniform spaces is called a uniform isomorphism; explicitly, it is a uniformly continuous bijection whose inverse is also uniformly continuous. A uniform embedding is an injective uniformly continuous map : between uniform spaces whose inverse : is also uniformly continuous, where the image () has the subspace uniformity ...
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
The continuous uniform distribution with parameters = and =, i.e. (,), is called the standard uniform distribution. One interesting property of the standard uniform distribution is that if u 1 {\displaystyle u_{1}} has a standard uniform distribution, then so does 1 − u 1 . {\displaystyle 1-u_{1}.}
An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is uniformly continuous; here, continuity is a local property of the function, and uniform continuity the corresponding global property.