# Uniform continuity

With uniformly continuous functions, a rectangle with height and width can be drawn around each point of the graph without the graph being directly above / below the rectangle. The function is uniformly continuous. Here the graph runs only within the rectangle. This is not the case with the function . With small arguments close to zero, the function changes so much that function values ​​are directly above or below the rectangle.${\ displaystyle 2 \ varepsilon}$${\ displaystyle 2 \ delta}$${\ displaystyle g (x) = {\ sqrt {x}}}$${\ displaystyle f (x) = {\ tfrac {1} {x}}}$

A uniformly continuous function is a term from the mathematical branch of analysis . Uniform continuity of a function is a stronger condition than that of the continuity of a function . In the case of a uniformly continuous function, the distance between any pairs of function values ​​is smaller than any given maximum error, as long as the arguments are sufficiently close to one another.

## definition

Be a subset of , short . ${\ displaystyle D}$${\ displaystyle \ mathbb {R}}$${\ displaystyle D \ subseteq \ mathbb {R}}$

A mapping is called uniformly continuous if and only if ${\ displaystyle f \ colon D \ rightarrow \ mathbb {R}}$

${\ displaystyle \ forall \ varepsilon> 0 ~ \ exists \ delta> 0 ~ \ forall x, x_ {0} \ in D \ colon | x-x_ {0} | <\ delta \ Rightarrow | f (x) -f (x_ {0}) | <\ varepsilon}$.

For a better differentiation, the usual continuity , if it is given in every point of , is also called point-wise continuity . ${\ displaystyle D}$

The peculiarity of the uniform continuity is that it only depends on and not, as with point-by-point continuity, additionally on the position . ${\ displaystyle \ delta}$${\ displaystyle \ varepsilon}$${\ displaystyle x_ {0}}$

This clearly means: For every vertical rectangle side , no matter how small , you can find a sufficiently small horizontal rectangle side so that if you move the rectangle with the sides along the function graph, it only cuts the vertical rectangle sides . (Ex .: root function open ). ${\ displaystyle \ varepsilon}$${\ displaystyle \ delta}$${\ displaystyle \ varepsilon; \ delta}$${\ displaystyle (0, \ infty)}$

### Examples

Consider the function

${\ displaystyle f \ colon \ mathbb {R} ^ {+} \ rightarrow \ mathbb {R} ^ {+}}$with :${\ displaystyle f (x) = x ^ {2}}$

This is continuous, but not evenly continuous: the further to the right you choose two points in one of the stripes, the greater the distance between the two function values ​​can be and thus exceed our selected one . This does not correspond to the definition of uniform continuity: the distance between the function values ​​must be smaller than a specified one for each choice of two such positions . This is not the case with this function. ${\ displaystyle \ delta}$${\ displaystyle \ varepsilon}$${\ displaystyle \ varepsilon}$

Furthermore, every restriction from to a compact interval is uniformly continuous. This follows directly from Heine's theorem . ${\ displaystyle f}$

Another example is the continuous function

${\ displaystyle f \ colon \ mathbb {R} ^ {+} \ rightarrow \ mathbb {R} ^ {+}}$ With ${\ displaystyle f (x) = {\ sqrt {x}}}$

### Generalization: metric spaces

The following definition is also used more generally:

Let be two metric spaces. A mapping is called uniformly continuous if and only if ${\ displaystyle (X, d_ {X}), (Y, d_ {Y})}$${\ displaystyle f \ colon X \ rightarrow Y}$

${\ displaystyle \ forall \ varepsilon> 0 ~ \ exists \ delta> 0 ~ \ forall x, x_ {0} \ in X \ colon d_ {X} (x, x_ {0}) <\ delta \ Rightarrow d_ {Y } (f (x), f (x_ {0})) <\ varepsilon}$.

### Generalization: uniform spaces

Even more generally, in topology, a function is called a function between two uniform spaces and uniformly continuous if the archetype of each neighborhood is a neighborhood again, i.e. if${\ displaystyle f \ colon X \ to Y}$ ${\ displaystyle (X, {\ mathcal {U}} _ {X})}$${\ displaystyle (Y, {\ mathcal {U}} _ {Y})}$${\ displaystyle (f \ times f) ^ {- 1} ({\ mathcal {U}} _ {Y}) \ subset {\ mathcal {U}} _ {X}.}$

## properties

Every uniformly continuous function is continuous. The reverse is not true: There are continuous functions like the square function that are not uniformly continuous. For certain domains of definition, continuity and uniform continuity coincide. The Heine theorem says: Every continuous function on a compact set is uniformly continuous.

If there is a Cauchy sequence in space and is uniformly continuous, then there is also a Cauchy sequence in . This generally does not apply to functions that are only continuous, as the example and shows. ${\ displaystyle (x_ {n}) _ {n \ in \ mathbb {N}}}$${\ displaystyle M}$${\ displaystyle f \ colon M \ to N}$${\ displaystyle (f (x_ {n})) _ {n \ in \ mathbb {N}}}$${\ displaystyle N}$${\ displaystyle M = (0,1], f (x) = {\ tfrac {1} {x}}}$${\ displaystyle x_ {n} = {\ tfrac {1} {n}}}$

Immediately from the fact that Cauchy sequences are mapped to Cauchy sequences, it now follows: If is uniformly continuous on a set , then it can be continuously continued on the closure . ${\ displaystyle f}$${\ displaystyle f}$${\ displaystyle M}$${\ displaystyle f}$ ${\ displaystyle {\ overline {M}}}$

In , the statement can clearly be made that a uniformly continuous function (with values ​​in ) cannot have any poles. How should it, since it can - as already shown - be continued steadily towards the end of its domain of definition. Such a continuous continuation is just not possible in a pole. ${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle \ mathbb {R}}$

Special forms of uniform continuity are Hölder and Lipschitz continuity .

## Visualization

In a uniformly continuous function for each predetermined maximum error an all pairs are found, then that function values and a maximum difference, as long as the distances from and smaller than shown. Accordingly, a rectangle with height and width can be drawn around each point of the graph , in which the graph runs completely inside the rectangle, so that no function values ​​are directly above or below the rectangle. This is not possible for functions that are not uniformly continuous. In part, the graph runs inside the rectangle - but not everywhere. ${\ displaystyle \ varepsilon> 0}$${\ displaystyle \ delta> 0}$${\ displaystyle f (x)}$${\ displaystyle f (y)}$${\ displaystyle \ varepsilon}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle \ delta}$${\ displaystyle (x, f (x))}$${\ displaystyle 2 \ varepsilon}$${\ displaystyle 2 \ delta}$