Basic definition of 1. function continuity
In mathematics, the condition that the function f(x) is continuous at a certain point x=a is:
F(a) exists (that is, the function is defined at this point);
Limx→af(x) exists;
limx→af(x)=f(a).
If a function satisfies the above conditions at every point in its domain, then the function is continuous in the whole domain.
2. Continuity of public elementary functions
Polynomial function: All polynomial functions are continuous within their definition domain.
Rational function: A rational function is continuous in its domain except the point where the denominator is zero.
Exponential function and logarithmic function: both exponential function and logarithmic function are continuous within their definition domain.
Trigonometric function: sine function, cosine function, tangent function, etc. Are continuous within its definition domain.
3. Properties of continuous functions
The sum, difference, product and quotient of continuous functions are still continuous functions: if the functions f(x) and g(x) are continuous in a certain interval, their sum, difference, product and quotient (if the denominator is not zero) are also continuous in that interval.
Continuity of compound function: If function g(x) is continuous at point A and function f(x) is continuous at point g(a), then compound function f(g(x)) is continuous at point X = A. ..
4. Judgment method of continuity
Judgment of point continuity: the function is continuous at a certain point x=a if and only if f(a) exists and limx→af(x)=f(a).
Judgment of interval continuity: If every point of a function is continuous in the interval [a, b], then the function is continuous in the interval.
5. Classification of common discontinuities
The first kind of discontinuous points: the left and right limits of the function exist at these points, but they are not equal.
Discontinuous points of the second kind: at these points, the limit of at least one direction of the function does not exist (or is infinite).
Jump discontinuity: there is a sudden change in the value of the function at these points.
The above are the basic judgment methods of continuity and some common rules. In practical application, when judging the continuity of a function, it is necessary to carefully observe the definition, image and properties of the function, and analyze it in combination with the above laws to determine whether the function is continuous at a certain point or interval. Different types of functions may need to be judged by different methods, so when learning and mastering the continuity of functions, we can better judge the continuity of functions and deepen our understanding of the essence of functions by comprehensively using these laws and methods.