Function and limit
6. Continuity of function: Let the function y=f(x) be defined in a neighborhood of point x0. If the limit of function f(x) exists at x→x0 and is equal to its function value f(x0) at point x0, that is, lim(x→x0)f(x)=f(x0), it is called function f(x0).
Discontinuity: 1, undefined at x=x0; 2. Although it is defined as x=x0, lim(x→x0)f(x) does not exist; 3. Although x=x0 is defined and lim(x→x0)f(x) exists, when lim(x→x0)f(x)≠f(x0), the function is said to be discontinuous or discontinuous at x0.
If x0 is a discontinuous point of the function f(x), but both the left limit and the right limit exist, it is called the first kind of discontinuous point of the function f(x) (if the left and right limits are equal, it is called a broken point; if not, it is called a jump discontinuity). Any discontinuity that is not the first discontinuity is called the second discontinuity (infinite discontinuity and oscillatory discontinuity).
Theorem The sum, product and quotient (denominator is not 0) of finite functions that are continuous at a certain point are functions that are continuous at that point.
Theorem If the function f(x) monotonically increases or decreases and is continuous in the interval Ix, then its inverse function x=f(y) monotonically increases or decreases and is continuous in the corresponding interval Iy={y|y=f(x), x∈Ix}. Inverse trigonometric functions are all continuous in their domain.
Theorems (Maximum Theorem and Minimum Theorem) A continuous function in a closed interval must have a maximum and a minimum in this interval. If the function is continuous in the open interval, or there are discontinuous points in the closed interval, then the function does not necessarily have a maximum value and a minimum value in this interval.
Theorem (boundedness theorem) A function that is continuous on a closed interval must be bounded on that interval, that is, m ≤ f(x) ≤ m Theorem (zero theorem) assumes that the function f(x) is continuous on a closed interval [a, b], and the signs of f(a) and f(b) are different (that is, f(a)×f(b).
From this, it can be inferred that the continuous function on the closed interval must obtain any value between the maximum value m and the minimum value m.
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