Advanced Mathematics ❤️ Chapter 1 ~ Section 3 ~ Limit ❤️ The concept of continuity

Functional continuity is a basic and important concept in advanced mathematics learning. It describes the smoothness and non-discontinuity of a function at a certain point, and is the cornerstone of establishing calculus theory and solving practical problems. This article will explain in depth the concept of function continuity, the types of continuity, the properties of continuous functions, and the application of function continuity in practical problems, leading readers to understand the charm of this important concept.

1. The concept of functional continuity

In mathematics, a function f(x) is continuous at a point x=a if and only if the following three conditions are simultaneously satisfied:

The function is defined at the point x=a, that is, f(a) exists;
The limit of the function at the point x=a exists, that is, lim(x→a) f(x) exists;
The limit of a function at point x=a is equal to the value of the function at that point, ie lim(x→a) f(x) = f(a).

The concept of function continuity shows that the function does not jump or break at a certain point, and its value can be obtained continuously around this point.

The type of continuity

There are three basic types of functional continuity:

The first type of continuity: the function is continuous at a certain point, which satisfies the definition of continuity.
The second type of continuity: the function is continuous in a certain interval, that is, every point of the function satisfies the definition of continuity.
The third type of continuity: the function is continuous in the entire domain, that is, every point of the function satisfies the definition of continuity.

3. Properties of continuous functions

Continuous functions have some important properties in mathematics:

The sum of two continuous functions is a continuous function. That is, if the functions f(x) and g(x) are continuous at a certain point x=a, then their sum [f(x) + g(x)] is also continuous at this point x=a.
The product of two continuous functions is a continuous function. That is, if the functions f(x) and g(x) are continuous at a certain point x=a, then their product [f(x) * g(x)] is also continuous at this point x=a.
A composite function of a continuous function is a continuous function. That is, if the function f(x) is continuous at the point x=a, and g(x) is continuous at the point x=f(a), then the composite function g[f(x)] is also continuous at the point x=a.

Necessary memory knowledge points

Knowledge point 1

Knowledge point 2

Examples (used to familiarize yourself with the continuity of functions in advanced mathematics)

Example 1

Example 2

Example 3

Example 4

Example 5

in conclusion

Functional continuity is a basic and important concept in advanced mathematics. It describes the smoothness and non-discontinuity of a function at a certain point, and is the cornerstone of establishing calculus theory and solving practical problems. The nature and application of function continuity make it widely used and important in mathematics and practical fields. Through a deep understanding of the concept and properties of function continuity, we can better understand the seamless link between mathematics and practice.