Uniform continuity – Serlo

Uniform continuity is a stronger form of continuity. It is derived from the epsilon-delta criterion of continuity and particularly important for approximating functions.

Motivation Bearbeiten

Recap: Epsilon-delta criterion Bearbeiten

The definition of uniform continuity is built upon that one of the epsilon-delta criterion for continuity. We will therefore recap the epsilon-delta definition:

A function   with   is continuous at  , if and only if for every   there is a   , such that   holds for all   with   .

That is, if we choose an epsilon-tube around   , no matter how small it is, all function values in a sufficiently small environment of   will lie in it. An epsilon-tube is an interval from   to   with   around :

 
A function f with an epsilon-tube around f(x_0)

A function   is continuous at   if and only if for every epsilon-tube, there is a   , such that all function values of   when evaluated between   and   fit inside the epsilon-tube:

 
All function values for x between x_0-delta and x_0+delta fit inside the epsilon-tube around x_0

This   may depend on the given function  , the given tube size  and from the point   . The following diagram shows an example, where a  -value is sufficiently small for   , but too large for   :

 
The value for delta is sufficiently small for x_0, but too large for x_1.

Hence, we must choose the   at   smaller. As both  -values at   and   are different, we will label them correspondingly by   and  . This shows, that the   within the definition of continuity may depend on the considered position  . The following diagram illustrates this:

 
Both the interval sizes delta_1 and delta_2 are sufficiently small for the given epsilon.

Derivation of uniform continuity Bearbeiten

The epsilon-delta criterion guarantees that any continuous function   can be approximated. For every maximal error   and around every   we can find a  , such that the function value   at every   inside the delta-interval   around   maximally deviates from   by  . So for all of those   with   we can consider   as an approximation of   with a maximal error of   . The following diagram illustrates this approximation at several points  :

 
Approximation of a continuous function at several positions.

However, the  -values depend on the considered position   . This is why the rectangles in the above diagram have different sizes. For a more uniform approximation, we could impose additionally that all rectangles have to be of equal size, That means, the  -value shall be equally suitable for any   . The above diagram would then look as follows:

 
Uniform approximation of a continuous function.

This is the core idea of uniform continuity. For any given   , there is a global  , such that no matter which point   is chosen, every function value   picked from the delta-interval   will be closer than   to   . This leads us to the following definition of uniform continuity for a function  , which will allow us for uniform approximations:

For every   there is a   (independent from the position  ), such that for every   and every argument   with   the following inequality holds:   .

Definition Bearbeiten

Definition of uniform continuity Bearbeiten

The formal definition of uniform continuity hence reads as follows:

Definition (Uniform continuity)

A function   is uniform continuous on D, if for every   a   exists, such that at all positions   and for all arguments   with a distance smaller than   to   the function values   and   deviate from each other by less than   . In quantifier notation, the definition of uniform continuity reads:

 

In different words, for each   there is a   , such that all pairs   with   satisfy the inequality   .

Quantifier notation Bearbeiten

The commented version of the quantifier notation reads:

 

Comparing the epsilon-delta definition of uniform continuity and continuity in the quantifier notation, we notide that only two quantifiers have been permuted:

 

The reason is that the   used for uniform continuity is a global one, i.e. it does not depend on the position   . In order to express this independence, the existence quantifier „ “ must appear in front of the all quantifier „ “ .

Deriving the negation of uniform continuity Bearbeiten

The quantifier notation is a handy tool for inverting definitions - like for uniform continuity. It basically works by step-wise replacing existence and all quantifiers by the respective other expression:

 

So:

 

Negation of uniform continuity Bearbeiten

Definition (Not uniformly continuous)

A function   is called not uniformly continuous, if there is at least one   , such that no matter how small   is chosen, we can find at least two arguments   and   with distance smaller than   , such that the function values   and   have a distance of at least   .

Uniform continuity is a global property Bearbeiten

Uniform continuity is a global property of a function. That means, it makes only sense to consider uniform continuity of a whole function. By contrast, continuity is a local property. We can speak of a function being continuous at a certain position  . For uniform continuity, this is not possible. However, a function can be both be globally continuous (continuous at every  ) and (globally) uniformly continuous.

The globalness is just a consequence from the definition: For uniform continuity, we can find a global  , whose delta-interval around every argument is a sufficient approximation. It is no necessary to speak of a global delta at some certain argument  : we just need to find one delta for all arguments.

Visualization Bearbeiten

Recap: visaulization of the epsilon-delta criterion Bearbeiten

In order to visualize the epsilon-delta criterion, we will draw a rectangle centred at   with height   and width  . In order to satisfy the epsilon-delta criterion, the graph must not lie above or below the rectangle, but completely inside:

 
The 2epsilon-2delta rectangle with permitted and non-permitted area

Take, for instance the square function at arguments around  . No matter how small we impose   , we can always find a  , such that the graph lies completely in the interior of the  - -rectangle:

 
Visualization of the epsilon-delta criterion for the square function

Conversely, if a function is not continuous, such as the sign function   , we are able to find an  , whose graph always takes values above or below the rectangle - no matter how narrow we make it. Concerning the sign function at   , this will for instance be the case if we choose   :

Visualization of uniform continuity Bearbeiten

 
For uniformly continuous functions, we can draw one rectangle with heighth   and width   and move it to any point on the graph, without having pieces of the graph above or below the rectangle. The function   is uniformly continuous. Despite having infinite derivative at zero, the graph always runs inside the upper and lower bounds of the rectangle. However, for   this is not the case. In the vicinity of arguments near zero, the function moves infinitely far into the  -direction. No matter how narrow we choose the rectangle, the function will always break its lower and upper barriers for   close enough to zero.

For uniform continuity,   has to be the same at every   . The graph has to fit inside the rectangle, no matter where we move it. That means: For every   there must be a   , such that the  - -rectangle can be moved arbitrarily along the graph without having function values above or below it:

 
For a uniformly continuous function, there is no point above or below the rectangle - no matter where on the graph we place it.

For a non-uniformly continuous function, this is not possible. Anoter counter-example is the square function. In fact, for any given   (not just a specific one), we cannot find a   , such that the graph does nowhere cross the upper or lower border of the  - -rectangle. For  -values near zero, the square function is close to constant, so the graph will fit inside some  - -rectangle, there. But the more we move it to the right, the steeper our function will get. And at some point, the function will break the upper and lower barriers of the  - -rectangle. So the square function in a continuous but not uniformly continuous function:

 
The square function is a continuous but not uniformly continuous function.

Scheme of proof Bearbeiten

Scheme of proof: Uniform continuity Bearbeiten

In terms of quantifiers, the definition of uniform continuity reads:

 

This can be used to derive a scheme of proof for uniform continuity:

 

Scheme of proof: Not uniformly continuous Bearbeiten

In terms of quantifiers, the definition of   being not uniformly continuous reads:

 

This can be used to derive a scheme for disproving uniform continuity:

 

Uniform continuity as a stronger form of continuity Bearbeiten

Uniform continuity is a stronger form of continuity. That means, every uniformly continuous function is also continuous. The converse does not hold. There are functions like the square function   being continuous, but not uniformly continuous:

 

Every uniformly continuous function is also continuous Bearbeiten

Theorem (uniform continuity implies continuity)

Every uniformly continuous function   with   is continuous at each point   .

Proof (uniform continuity implies continuity)

We need to show that a uniformly continuous function   is continuous at every point   . That means, we are given an arbitrary   and   and we need to find a   such that   implies the inequality   . But now, we can just apply the definition of uniform continuity. This definition provides us with a  , such that at all   with   , there is  . As this implication holds for all   , it particularly holds for   and  . So the   given by uniform continuity can also be used as the   for ordinary continuity at  .

The square function is continuous but not uniformly continuous Bearbeiten

As we have seen, every uniformly continuous function is also continuous. However, the converse does not hold. Let us consider again the square function as an example:

 

As we have seen in the section visualization, if there is   given, we cannot set a fixed   , such that the graph is everywhere inside the  - -rectangle , independent from where we place it. The further we move the rectangle to the right, the steeper the square function gets. And at some point, the graph starts crossing he upper and lower borders of the  - -rectangle.

This can also be seen by checking the epsilon-delta definition of uniform continuity:  . We want to prove the negation of this statement, i.e.   and  . Consider for instance   and assume there was some  , such that   for all real numbers   with   . Now consider some   which we will determine later. We take the centre   and some point   inside the  -interval. That means,  .

Now, we want to show that   holds.

 

Further,

 

If we choose   then it is guaranteed that  . Hence, we have proven that

 

is not uniformly continuous.

Example: uniformly continuous functions Bearbeiten

  • The identity function   is uniformly continuous, since for   , we can show   by choosing   .
  • Above, we have seen that the square function   is not uniformly continuous on the real numbers. However, when restricting to a compact interval, the function gets uniformly continuous. For example,
 

is uniformly continuous. We can prove this as follows: There is

 

because   . So we can choose   and then for all   with   the estimate   will hold.

  • The square root function is uniformly continuous on  . Consider:
 

Let   be arbitrary. Then,   is a suitable choice: Let   with  .

Without any restrictions, we may choose  . Then, there is also  , and hence we get  .

Now, we want to prove that   .

By assumption there is  . And therefore:

 

So  . Since by assumption   and also   , we may take the square root of this equation and obtain  , i.e. , which is what we wanted to show. Hence, we have proven that   is uniformly continuous.

  • The following example is not uniformly continuous:
 

, which are sine waves oscillating faster and faster when approaching  . Assume that   was uniformly continuous. Then we could find some global   such that the epsilon-delta criterion holds everywhere. But now, for   the frequency of   gets arbitrarily high, so inside an interval close to   , there will always be a full oscillation of the sine inside any  -ball. Since the peak-to-peak amplitude of the sine function is equal to   , the condition   will never hold everywhere for   . We just have to move the rectangle close enough to 0, as illustrated by the subsequent figure:

 
This image shows, that the topological sine curve is not uniformly continuous. For an environment of fixed width and x going towards 0, at some point a full period will be inside the environment.

Properties Bearbeiten

As we have seen, not every continuous function is also uniformly continuous. But if we restrict to a closed, compact interval   , both continuity and uniform continuity will be equivalent:

Theorem (Heine-Cantor for  )

Every function   is uniformly continuous.

Proof (Heine-Cantor for  )

We choose an indirect way of proof: suppose, the function   was not uniformly continuous. That means, there is an   and for every   there are two points  , such that   but  .

The Bolzano Weierstraß theorem tells us (this is where compactness of   comes into play) that the bounded sequence   must have a convergent subsequence  , whose limit   is inside the interval   . Since   this is also the limit of the subsequence  .

Now, continuity of   implies   and  . Therefor, there must be a  , such that   and   for all  . We proceed by splitting and bounding   ,which holds for all   . This contradicts our assumption   for all  . The assumption of   not being uniformly continuous must therefore have been wrong and we obtain uniform continuity.