Uniform continuity

The concept relies on comparing the sizes of neighbourhoods of distinct points, so it requires a metric space, or more generally a uniform space.

, and this can be determined by looking at only the values of the function in an arbitrarily small neighbourhood of that point.

, in the sense that the standard definition of uniform continuity refers to every point of

On the other hand, it is possible to give a definition that is local in terms of the natural extension

(the characteristics of which at nonstandard points are determined by the global properties of

), although it is not possible to give a local definition of uniform continuity for an arbitrary hyperreal-valued function, see below.

) is expressed by a formula starting with quantifications (metrics

) are rotated: Thus for continuity on the interval, one takes an arbitrary point

, uniform continuity requires the existence of a positive real number

This means that there is no specifiable (no matter how small it is) positive real number

However, the image of a bounded subset of an arbitrary metric space under a uniformly continuous function need not be bounded: as a counterexample, consider the identity function from the integers endowed with the discrete metric to the integers endowed with the usual Euclidean metric.

The Darboux integrability of continuous functions follows almost immediately from this theorem.

For a uniformly continuous function, for every positive real number

of the graph, if we draw a rectangle with a height slightly less than

around that point, then the graph lies completely within the height of the rectangle, i.e., the graph do not pass through the top or the bottom side of the rectangle.

(the graph penetrates the top or bottom side of the rectangle.)

The proofs are almost verbatim given by Dirichlet in his lectures on definite integrals in 1854.

Uniform continuity can be expressed as the condition that (the natural extension of)

, but at all points in its non-standard counterpart (natural extension)

, and this has the further pleasant consequence that if the extension exists, it is unique.

It is easy to see that every uniformly continuous function is Cauchy-continuous and thus extends to

In general, for functions defined on unbounded spaces like

(assuming the existence of qth roots of positive real numbers, an application of the Intermediate Value Theorem).

A typical application of the extendability of a uniformly continuous function is the proof of the inverse Fourier transformation formula.

We first prove that the formula is true for test functions, there are densely many of them.

In the special case of two topological vector spaces

This fact is frequently used implicitly in functional analysis to extend a linear map off a dense subspace of a Banach space.

Just as the most natural and general setting for continuity is topological spaces, the most natural and general setting for the study of uniform continuity are the uniform spaces.

Each compact Hausdorff space possesses exactly one uniform structure compatible with the topology.

A consequence is a generalization of the Heine-Cantor theorem: each continuous function from a compact Hausdorff space to a uniform space is uniformly continuous.

As the center of the blue window, with real height $2\varepsilon \in \mathbb {R} _{>0}$ and real width $2\delta \in \mathbb {R} _{>0}$ , moves over the graph of $f(x)={\tfrac {1}{x}}$ in the direction of $x=0$ , there comes a point at which the graph of $f$ penetrates the (interior of the) top and/or bottom of that window. This means that $f$ ranges over an interval larger than or equal to $\varepsilon$ over an $x$ -interval smaller than $\delta$ . If there existed a window whereof top and/or bottom is never penetrated by the graph of $f$ as the window moves along it over its domain, then that window's width would need to be infinitesimally small (nonreal), meaning that $f(x)$ is **not** uniformly continuous. The function $g(x)={\sqrt {x}}$ , on the other hand, is uniformly continuous.