The concept of "density" often appears in functional analysis and real variable function, which is used to measure the inclusion relationship between two sets: let (x, p) be a metric space, set E be a subset of X, if X is any element of X, any positive number epss & gt0, and element Z is in E, so p (z, x).
The concept of continuity first appeared in mathematical analysis, and then it was extended to point set topology.
Suppose f: x-> Y is the mapping between topological spaces, and if the following conditions are met, it is said that F is continuous: for any open set U on Y, the original image F (- 1) (U) of U under F must be an open set on X.
For any point in an interval, it has its own definition, and its left limit and right limit are equal and both exist, then the function is said to be continuous in this interval.
definition
If f(x) is defined in a neighborhood U(x0) of x0, and the limit of f(x) is f(x0) when x approaches x0, the function f(x) is said to be continuous at x=x0.
Consistent and continuous: 1. It is known that if the function f(x) defined on the interval I is for any real number b >；; 0, there is a real number c>0 that makes x 1, x2 and x 1, x2 satisfy |x 1-x2| on any I.
2。 If a function is continuous in the closed interval [a, b], it is uniformly continuous in the closed interval [a, b].
Continuity is relative to discontinuity, and the whole world is made up of these two things, such as light. At present, it has continuity and discontinuity. Many methods of mathematics also extend from discontinuous to continuous, such as calculus. Continuity means that discontinuity approaches it infinitely, which forms continuity. At present, the difference between the two is still very vague.