Uniform continuity

Uniform continuity is a term from the analysis. He referred to a special case of continuity.

Definition

Is a subset of is short.

A mapping is called uniformly continuous if and only if

To better distinguish the normal consistency is referred to when it is placed in any point of, also known as point-wise consistency.

The peculiarity of the uniform continuity is that not only depends on and as with the pointwise continuity, additionally from the spot.

Clearly, this means that to every small vertical side of the rectangle, you can find a sufficiently small horizontal side of the rectangle, so if you suitable skirting the rectangle with the sides on the function graph, it always cuts only the vertical sides of the rectangle. (Eg: root function on ).

Examples

Consider the function

This is continuous but not uniformly continuous, the further right you at a distance less than two selected points, the greater the distance between the two function values ​​. This does not meet the definition of uniform continuity: the distance of the function values ​​must be less than a given for every choice of two such places to be. This is not the case with this function.

Furthermore: Any restriction of a compact interval is uniformly continuous. This follows immediately from the theorem of Heine.

Another example is the continuous function

Which is uniformly continuous, even Holder continuous, but not Lipschitz continuous.

Generalization: metric spaces

More generally the following definition is used:

Be two metric spaces. A mapping is called uniformly continuous if and only if

Generalization: uniform spaces

More generally, the topology is called a function between two uniform spaces and uniformly continuous if the inverse image of each neighborhood is a neighborhood again, if so

Properties

It applies: If uniformly continuous on a set, then is also continuous at every point and even continuously continuable on the financial statements. Conversely, however, there are continuous functions which are not uniformly continuous.

A simple criterion for the detection of uniform continuity is the set of Heine: Every continuous function on a compact set is uniformly continuous.

Also a Cauchy sequence is a Cauchy sequence in the space and is uniformly continuous, so is. This does not apply to features that are only ever as the example and shows in general.

In: poles can not exist on a uniformly continuous function, since at infinity aspiring slope of the distance of the function values ​​is arbitrarily large, so no real can exist.

Special forms of uniform continuity are Hölder and Lipschitz continuity.