Uniform continuity

In mathematics, uniform continuity is a special way that some functions behave. A function is uniformly continuous if its values don't change too much over small distances. This means that for any small distance between function values, you can find an even smaller distance in the inputs to keep the outputs close.

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 small input distance needed to keep outputs close can change depending on where you are. With uniform continuity, one small input distance works everywhere. This makes uniformly continuous functions very smooth and easy to predict.

Uniformly continuous functions are useful in analysis and many areas of mathematics. They can be described using something called a modulus of continuity, which measures how smoothly the function changes.

History

In 1870, a mathematician named Heine first talked about uniform continuity. Two years later, he showed that a function that changes smoothly on an open stretch might not always be uniformly continuous. Before that, another mathematician named Bolzano had also worked with this idea. He showed that continuous functions on a closed interval are uniformly continuous, but he did not fully prove it.

Other characterizations

Non-standard analysis

In non-standard analysis, a function is microcontinuous at a point if its values change only a tiny amount when the input changes a tiny amount. This means a function is continuous on a set if it is microcontinuous at every point in that set. Uniform continuity means the function is microcontinuous at all points, including points that are not real numbers. Note that some functions follow this rule but are not uniformly continuous, and some uniformly continuous functions do not follow this rule exactly. For more details, see non-standard calculus.

Characterization via sequences

For a function between Euclidean spaces, uniform continuity can be explained using sequences. If a set is part of R n and a function is uniformly continuous, then for every two sequences where the points get very close to each other, the function values at those points also get very close to each other.

Relations with the extension problem

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

A function that is uniformly continuous is always Cauchy-continuous, so it can be extended. But 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 area B around zero in W, there is a matching area 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 straight line changes between these spaces, uniform continuity is the same as regular continuity. This idea is often used in functional analysis to extend a straight line map from a full area 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.