Uniform continuity

In mathematics, uniform continuity is a special way that some functions behave. A function is uniformly continuous if, no matter where you look in its domain, the values of the function don't jump around too much over small intervals. This means that for any tiny distance you pick between function values, you can find an even tinier distance in the input values that keeps the output within that chosen range.

As the center of the blue window, with real height 2 ε ∈ R > 0 {\displaystyle 2\varepsilon \in \mathbb {R} _{>0}} and real width 2 δ ∈ R > 0 {\displaystyle 2\delta \in \mathbb {R} _{>0}} , moves over the graph of f ( x ) = 1 x {\displaystyle f(x)={\tfrac {1}{x}}} in the direction of x = 0 {\displaystyle x=0} , there comes a point at which the graph of f {\displaystyle f} penetrates the (interior of the) top and/or bottom of that window. This means that f {\displaystyle f} ranges over an interval larger than or equal to ε {\displaystyle \varepsilon } over an x {\displaystyle x} -interval smaller than δ {\displaystyle \delta } . If there existed a window whereof top and/or bottom is never penetrated by the graph of f {\displaystyle 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 ) {\displaystyle f(x)} is not uniformly continuous. The function g ( x ) = x {\displaystyle g(x)={\sqrt {x}}} , on the other hand, is uniformly continuous.

Uniform continuity is stronger than regular continuity. With regular continuity, the tiny input distance needed to keep outputs close can depend on where you are in the function. With uniform continuity, one tiny input distance works everywhere. This makes uniformly continuous functions very predictable and smooth.

Uniformly continuous functions have useful properties in analysis and are important in many areas of mathematics. They can be described using something called a modulus of continuity, which gives a precise way to measure how smoothly the function changes.

History

In 1870, a mathematician named Heine first described the idea of uniform continuity. Two years later, he showed that a function that changes smoothly on an open stretch can still behave in ways that aren't uniformly continuous. Earlier, another mathematician named Bolzano had also worked with this idea and showed that continuous functions on a closed interval are uniformly continuous, though he did not fully prove it.

Other characterizations

Non-standard analysis

In non-standard analysis, a real-valued function f is microcontinuous at a point a precisely if the difference f ∗( a + δ ) − f ∗( a ) is very small whenever δ is very small. This means f is continuous on a set A if f ∗ is microcontinuous at every real point in A. Uniform continuity can be described by saying that f is microcontinuous not only at real points in A but also at all points in its non-standard version ∗A. Note that there are special functions that fit this rule but aren’t uniformly continuous, and some uniformly continuous functions don’t fit this rule. For more details, see non-standard calculus.

Characterization via sequences

For a function between Euclidean spaces, uniform continuity can be described using sequences. If A is a part of R n and a function f: A → R n is uniformly continuous, then for every pair of sequences x n and y n where the distance between x n and y n gets very small, the distance between f( x n ) and f( y n ) also gets very small.

Relations with the extension problem

When can a function be extended to work on a bigger space? If a special set is closed, we can use a special rule to extend the function. For this to work, the function needs to have a property called "Cauchy-continuous".

A function that is uniformly continuous is always Cauchy-continuous, which means it can be extended. However, not all Cauchy-continuous functions are uniformly continuous. For example, the function that squares its input is continuous and Cauchy-continuous but not uniformly continuous.

For functions that work on very large spaces, uniform continuity is a strong condition. Sometimes, we can use a weaker condition to extend the function. If a function is uniformly continuous on every bounded part of its space, it can be extended to the whole space.

One common use of extending uniformly continuous functions is in proving important math formulas. By showing the formula works for many special cases and then extending it, we can prove it works everywhere.

Generalization to topological vector spaces

In some special cases, we can talk about uniform continuity in more complex spaces called topological vector spaces. For two of these spaces, called V and W, a map f from V to W is uniformly continuous if, for any small neighborhood B around zero in W, there is a matching neighborhood A around zero in V. This means that if two points v₁ and v₂ in V are close together (within A), then their images f(v₁) and f(v₂) in W will also be close together (within B).

For linear transformations between these spaces, uniform continuity is the same as regular continuity. This idea is often used in functional analysis to extend a linear map from a dense subspace of a Banach space.

Generalization to uniform spaces

Just as the idea of continuity can be studied in special kinds of spaces called topological spaces, uniform continuity is best studied in structures called uniform spaces. In these spaces, a function is uniformly continuous if it keeps points that are close together in the input also close together in the output, no matter where you look in the space.

In this setting, uniformly continuous maps also have a nice property: they turn sequences that stay close together into sequences that stay close together in the output. Additionally, special spaces called compact Hausdorff spaces have a unique way to support this idea of uniform closeness, and this leads to the result that every continuous function from such a space to a uniform space is automatically uniformly continuous.