The concept of continuity in advanced mathematics is: let the function y=f(x) be defined in a neighborhood of point x0. If the change of the independent variable △x approaches zero and the change of the corresponding function △y approaches zero, it is said that y=f(x) is continuous at point x0.

The function f(x) is continuous at the point x0, and the condition to be satisfied is: 1, and the function is defined at this point. 2. The limit lim(x→x0)f(x)=f(x0) of the function exists at this point. 3. The limit value is equal to the function value f(x0).

Function y=f(x) When the change of independent variable x is small, the change of dependent variable y is also small. For example, the temperature changes with time, as long as the time change is small, the temperature change is small; For another example, the displacement of a free-falling body changes with time. As long as the time change is short enough, the displacement change is also small.

For this phenomenon, we say that the dependent variable changes continuously about the independent variable, and the image of the continuous function in the rectangular coordinate system is a continuous curve without fracture. According to the nature of limit, the necessary and sufficient condition for a function to be continuous at a certain point is that it is continuous near that point.

For continuity, there are many phenomena in nature, such as temperature changes and plant growth, which are constantly changing. The reflection of this phenomenon in the function relationship is the continuity of the function.