continuity

Mathematical noun

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stay mathematics Medium, continuity yes function An attribute of. Intuitively speaking, a continuous function is a function whose output change will be small enough when the change of input value is small enough. If a slight change in the input value will cause a sudden jump in the output value or even can not be defined, this function is called Discontinuity Function of (or with Discontinuity ）。

The most fundamental definition of continuity is topology Definition in the entry Continuous function (topology) It will be discussed in detail in. stay Order theory especially Domain theory There is another abstract continuity derived from this basic concept: Scott continuity 。

Chinese name: continuity
Foreign name: Continuity

Appear earliest: mathematical analysis
Promotion: Point set topology

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four Related Item

Real valued continuous function

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The most basic and common continuous functions are Define Fields by real number A subset and value of a set are also real number Continuous function of. The continuity of the function can be used Rectangular coordinate system In image To represent. Such a function is continuous, if roughly speaking, its image is a single unbreakable curve, and there is no interrupted 、 jump or Oscillation of infinite approximation 。

Strictly speaking, set

Is a subset of the set of real numbers

Shoot to

Function of:

。

stay

A point in

Is continuous if and only if the following two conditions are met:

one

At point

There are definitions on.

two

yes

One of Accumulation point , and regardless of the argument

stay

How to approach

，

Of limit Both exist and equal to

。

We call function Continuous everywhere or Continuous everywhere , or simply called continuity , if it is continuous at any point in its domain. More generally, when a function is continuous at every point of a subset in the domain, it is said that the function is continuous on this subset.

definition

Instead of the concept of limit, the following so-called

Method to define the continuity of a real valued function.

Still consider function

。 hypothesis

yes

Element in the domain of the. function

Is known to be

The points are continuous if and only if the following conditions are true:

For any positive real number

, there is a positive real number

Make it possible to

, as long as

satisfy

, there is

Establishment.

Continuous“

Defined by Cauchy First, give.

More intuitively, the function

Is continuous if and only if any one is taken

Point in

Neighborhood of

, can be defined in its domain

Middle Pick Point

Is small enough to make

Neighborhood in function

The mapping on the will fall on

Neighborhood of

Within.

The above is for single variable functions (defined in

This definition is also true when it is extended to multivariable functions. metric space as well as topological space See the next section for the definition of continuous functions between.^[1]

example

All polynomial All functions are continuous. All kinds Elementary function , such as exponential function 、 Logarithmic function 、 Square root function And trigonometric function They are also continuous functions in their domain.
absolute value The function is also continuous.
Defined on non-zero real numbers reciprocal The function f=1/x is continuous. However, if the domain of a function is expanded to all real numbers, then no matter what value the function takes at zero, the expanded function is not continuous.
An example of a discontinuous function is a function defined piecewise. For example, f is defined as: f (x)=1 if x>0, f (x)=0 if x ≤ 0. If ε=1/2, there is no δ - neighborhood where x=0, so that all values of f (x) are within the ε neighborhood of f (0). Intuitively, we can regard this discontinuity as a sudden jump of function value.
Another example of a discontinuous function is Symbolic function 。

Properties of continuous function

If two functions f and g are continuous,

Is a real number, then

、

and

All are continuous. The set of all continuous functions forms a ring and also a vector space (actually constitutes a Algebra ）。 If all

, both

, then

It is also continuous. Composite function of two continuous functions

It is also a continuous function.

If the real function f is continuous in a closed interval, and

Is some

and

If there is a number between

Internal

, making

This theorem is called Intermediate value theorem 。 For example, if a child's height increases from one meter to 1.5 meters between the ages of five and ten, then there must be a time when his height is exactly 1.3 meters.

If f

Internally continuous, and

and

One positive and one negative, there must be a point in the middle

, making

。 This is a corollary of the intermediate value theorem.

If f is in the closed interval

If it is internally continuous, it must reach the maximum value, that is, there is always

, so that for all

, Yes

。 Similarly, a function must have a minimum value. This theorem is called Extremum theorem 。 (Note that if the function is defined in an open interval