Continuous function

The continuity is a concept of mathematics, which is mainly in the sub-areas of analysis and topology of central importance. A function is called continuous if sufficiently small changes of the argument ( the arguments) to arbitrarily small changes in the function value. This means in particular that no jumps occur in the function values ​​. If jumps only in one direction, it is called semi-continuity.

• 2.1 Concatenation of continuous functions
• 2.2 sums of continuous functions
• 2.3 continuity of the inverse function
• 2.4 Intermediate Value Theorem
• 2.5 theorem of Bolzano
• 2.6 Set the minimum and maximum

Definitions

The idea of continuity can be described as follows: A real-valued function on a real interval is continuous if the graph of the function without having the pin can be drawn. The function may especially have no discontinuities. This statement is not a definition, because on the one hand, it is unclear how to draw without lifting the pen could be expressed in mathematical terms. On the other hand, there are both continuous functions whose graphs exhibit jumps (example), as well as discontinuous functions whose graphs are no jumps in the intuitive sense have (eg Dirichlet function: two parallel solid lines). Nevertheless, it corresponds roughly to the importance of the stability and is therefore very useful for the intuition.

Augustin- Louis Cauchy and Bernard Bolzano gave the early 19th century independently a definition of continuity. They called a function continuous, if sufficiently WOULD CHOOSE small changes in the argument only to arbitrarily small changes in the function value by itself. This was already an exact definition, but in its practical application leaves open certain questions. The nowadays usual ε - δ criterion was introduced by Karl Weierstrass at the end of the 19th century.

It says in words as: The function is continuous at a point when it comes to any environment its image point is a neighborhood of which is imaged throughout the area.

Continuity of real functions

For real functions - that is functions whose domain and target range are subsets of the real numbers - there are several equivalent definitions of continuity usual:

Intuitively, this means that you can include all function values ​​in even the smallest environment, if one chooses the environment for the values ​​small enough.

An equivalent formulation used limits of sequences:

In short: always follows.

The concept of continuity of a function can be defined with the help of the notion of limit of a function:

An equivalent characterization of continuity is a topological:

In general, a function is continuous on if it is continuous at every point of its domain of definition.

Examples

• The sine function is continuous in.
• The cosine function is continuous in.
• (as a composition of exponential and the cosine function ) in steadily.
• The tangent function is continuous throughout its domain of definition. This arises due to, ie. Note: No points of discontinuity are the arguments, and the argument of the inverse function, since the functions are not defined at these points and continuity can only refer to points of the domain.
• The inverse function is continuous on its entire domain of definition.
• The sign function is continuous at each point, but discontinuous at the point 0: left-side limit is -1, the right side and thus, the threshold limit value 1 does not exist. Therefore, the sign function is not continuous at all.
• The function is continuous at 0 discontinuous (so-called Oszillationsstelle ), in all other respects.
• The Dirichlet function is discontinuous at each location.
• The thoma ash function on the interval is continuous at every rational point and discontinuous at every irrational point.
• Each function is continuous at every isolated point of its domain. In particular consequences are continuous in.

Properties

• Are and steadily on a common domain of definition, so are, , and steady; However, it must define the area of the case that one or a plurality of zeros has to be limited to the range.

• The composition of two continuous functions is also continuous.

Left-hand / right- continuity

A defined on a set function is left -continuous at a point when the left-sided limit exists and is. Is on the whole domain of left -continuous, so we can also say is left continuous. Analogously one defines right- continuity on the right-hand limit with analogous notation.

A defined on a subset of the real numbers is a function that is continuous at if exist in right - and left-hand limits and are. This allows for a classification of discontinuities.

Continuous supplementability

If a point, but an accumulation point of, so it may be that the ( two-sided ) limit exists. The added function and is then steadily in place. We say that the function is continuous at the point supplemented, and often used for the new function, the original designation. The supplementary value can be determined in many cases differentiable by the rule of L' Hospital.

Examples:

Generalization: Continuous functions between metric spaces

A function is called continuous if the function value sufficiently little change as long as it only changes the function argument enough little. The terms of the metric space can this description be formalized in various ways. and each metric spaces with the associated metrics, a function with domain of definition. The following definitions are equivalent:

In many topics of analysis are continuous maps between metric spaces into consideration. The function

Is, for example, continuously. Here are at fixed and fixed continuous functions. However, this is generally not a sufficient criterion for the continuity of. A counter-example is

This function is discontinuous at the point are even and, for each continuous or functions of a real variable.

Other relevant classes of continuous functions are the constant functions. The complex exponential function is an example of such a function.

Further generalization: Continuous functions between topological spaces

All previous definitions are specializations of the corresponding definition of continuity in the topology. There is a function between two topological spaces is said to be continuous if the inverse images of open sets are again open sets. A function is called sequentially continuous if it satisfies the follow- criterion, ie if

Applies to any convergent sequence of items.

Every continuous function is sequentially continuous. In rooms that satisfy the first axiom of countability, ie in particular in metric spaces, the converse also holds that each follow continuous function is continuous.

Order theoretical concept of continuity

Order Theoretically, the continuity conceived as a function of compatibility with the supremum complete posets. A function is called continuous if for all directed subsets. This concept plays a central role in the field theory. Similarly, the sequential continuity above are also depicted here again limits to limits.

In this context, it follows from the continuity of a function whose monotony. Conversely, every monotone function forms a directed set back on such as, making the existence of the supremum of the image then right from the start and is certainly no longer needs to be shown. Many authors take the monotony as a condition in the definition of continuity.

Other continuity terms

Tightening of the concept of continuity are, eg, uniform continuity, (local) Lipschitz continuity, Hölder continuity and absolute continuity. The usual continuity is sometimes referred to as pointwise continuity, to distinguish them against the uniform continuity. Applications the Lipschitz continuity can be found eg in uniqueness theorems (eg set of Picard - Lindelöf ) for initial value problems and in the geometric measure theory. The absolute continuity is used in the stochastic theory and measure theory.

A feature that may have a number of functions, the continuity gleichgradige. It plays a role in commonly used set of Arzelà - Ascoli.

Context

The following relationships apply in the case of real functions:

Lipschitz continuous locally Lipschitz continuous

And

Lipschitz continuous absolutely continuous uniformly continuous steady.

Examples

Some counter-examples are intended to demonstrate that the return directions do not apply in most cases:

• Is locally Lipschitz continuous, but not Lipschitz continuous or uniformly continuous.

• Is continuous but not locally Lipschitz continuous at the origin.

Important theorems on continuous functions

Concatenation of continuous functions

Each manifold (including composition, sequential execution or cascading called ) of continuous functions is again continuous.

Sums of continuous functions

Finite sums of continuous functions are continuous.

However, a number can be as a limit of a sequence of continuous functions even discontinuous, if it converges at each point to a finite limit. The oldest example of this is given in 1826 by Niels Henrik Abel series

Which is discontinuous at the location among others. However, are stronger requirements, such as the uniform convergence of the partial sums of the series, so the limit function is necessarily continuous.

Continuity of the inverse function

Are in an interval and a continuous, strictly increasing or strictly decreasing function, then the image of an interval is bijective, and the inverse function is continuous. Thus, it is a homeomorphism from to.

This applies as described for functions which are continuous in the entire interval. Is a reversible and continuous at the location function, the inverse function is not continuous at the point in general. As a counter- example is defined by:

Then is bijective and continuous at the point, but is discontinuous at.

In general, the following sentence is true: when is compact and Hausdorff, then the inverse function is continuous for every continuous bijection.

Intermediate value theorem

The intermediate value theorem states that a on the interval ( with ) a continuous function takes on every value between at least once.

Formal:

An equivalent formulation is: The image of a continuous function on an interval is again an interval. ( However, the image of an open or semi-open interval can be quite a closed interval. )

Theorem of Bolzano

As a special case of the intermediate value theorem there is a sentence of Bernard Bolzano: If the constant on a closed interval function in two places and this interval function values ​​with different signs, so there is between and at least one point at which the function vanishes, ie. The function has so there is a zero.

Set by the minimum and maximum

A real-valued function (the complete and limited ) to a compact subset of is continuously is limited, and assumes its upper and lower limit. For real functions which can be rephrased as follows: If is continuous, then there are places so that

Applies.

This proved by Weierstrass theorem, sometimes called the extreme value theorem, provides only the existence of these extreme values ​​. Statements from the differential calculus are useful for practical locating these points.

The statement is also valid on compact topological spaces.

Differentiability of continuous functions

Continuous functions are not necessarily differentiable. At the beginning of the 19th century was convinced that a continuous function at most a few places could not be differentiable (like the absolute value function ). Bernard Bolzano constructed it as the first mathematician a function that is continuous everywhere but nowhere differentiable, the Bolzanofunktion what was in the professional world, however, not known; Karl Weierstrass was then in the 1860s also such, known as the Weierstrass function function, which this time among mathematicians made ​​waves. Its function is defined as follows:

With an odd natural number and with.

Function spaces of continuous functions

The space of continuous real-valued functions on a topological space is a real vector space, it is called with. In this area in particular all differentiable functions are included, if an open subset of, or a differentiable manifold is. Functions whose derivatives are also continuous, it is called continuously differentiable. These features also form a linear space, which is called. Accordingly, we defined as the space of functions that are twice differentiable, where the- th derivative is continuous, which are so - times continuously differentiable. Furthermore, denotes the space of infinitely differentiable functions.