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 ${\displaystyle f(x)={\sqrt {x}}}$

which is evenly steady, even Hölder steady , but not lipschitz steady .

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. ${\displaystyle \varepsilon >0}$ ${\displaystyle \delta >0}$ ${\displaystyle f(x)}$ ${\displaystyle f(y)}$ ${\displaystyle \varepsilon }$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \delta }$ ${\displaystyle (x,f(x))}$ ${\displaystyle 2\varepsilon }$ ${\displaystyle 2\delta }$

• 

  With an evenly continuous function, a rectangle with height and width can be drawn for each point around each point , with the graph running completely inside the rectangle. ${\displaystyle \varepsilon >0}$ ${\displaystyle 2\varepsilon }$ ${\displaystyle 2\delta }$

• 

  If the function is not uniformly continuous, there are values for which it is impossible to find one, so that the graph runs everywhere inside the - rectangle. If you move the rectangle along the graph, there are places where function values ​​are directly above or below. ${\displaystyle \varepsilon >0}$ ${\displaystyle \delta >0}$ ${\displaystyle 2\varepsilon }$ ${\displaystyle 2\delta }$

See also

• Uniform uniformity

Web links

Wikibooks: Math for non-freaks: Uniform consistency  - learning and teaching materials

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• Konrad Königsberger : Analysis 1 . Springer-Verlag, Berlin et al., 2004, ISBN 3-540-41282-4 .
• Konrad Königsberger : Analysis 2. Springer-Verlag, Berlin / Heidelberg, 2000, ISBN 3-540-43580-8 .
• Otto Forster : Analysis Volume 1: Differential and integral calculus of a variable. Vieweg-Verlag, 8th edition 2006, ISBN 3-528-67224-2 .