The set of positive integers is denoted as N

The set of integers is denoted Z

The set of rational numbers is denoted Q

The set of real numbers is denoted R

The set of all complex numbers is denoted C

mapping

compound mapping

inverse mapping

summary

1,x>o

0,x=0

-1,x<0

1,x∈Q

0,x∈R-Q

Suppose the domain of the function y=f(x) is D. If there is a positive number M, so that for any point x in a certain sub-interval of D, there is always |f(x)|＜M (that is, -M＜f( x)<M), then the function f(x) is said to be a bounded function on D, otherwise it is an unbounded function.

Whether a function is bounded or unbounded must indicate the interval considered, because the same function may be bounded on one interval but unbounded on another interval.

Only when there is both an upper bound and a lower bound can it be called bounded (the world is not unique)

If f(-x)=f(x) is satisfied, it is an even function

If f(-x)=-f(x) is satisfied, it is an odd function

Judging parity after the four arithmetic operations of the parity function can be based on the four arithmetic operations of negative numbers.

The Dirichlet function has no minimum positive period

The essence of a composite function is to decompose a function into several simpler functions

Pay attention to the domain of the decomposed simple function and the original composite function

Assume A and B are sets of real numbers. The inverse mapping f -1→ of mapping f : A → B is called the inverse function of y = f (x), that is: if for each y∈B, there is a unique x∈A, Let y = f (x), then x is also a function of y, denoted as f -1, that is, x = f -1( y )

x = f -1( y ) is the inverse function of the function y = f ( x ); y = f ( x ) is the direct function of the inverse function x = f-1 ( y )

The inverse function is consistent with the monotonicity of the direct function (strictly monotonic increasing and decreasing)

Not all functions have inverse functions

There must be an inverse function for injective and one-to-one mapping

Generally use substitution when solving problems

Four steps for finding limits by definition method

There is a limit - convergence

No limit - divergence

If the sequence converges, then its limit is unique

Convergence must be bounded (all bounded)

Bounded does not necessarily mean convergence

If lim(n→infinity)=a, a>0 (or a<0), then N>0 exists. When n>N, Xn>0 (or Xn<0).

It can be generally understood that if the limit value is greater than 0, then Xn after N are all greater than 0.

If the sequence converges, then its subsequences must converge.

If both odd and even subsequences converge, then the sequence converges

A divergent sequence may also have convergent subsequences

Supplementary knowledge points

If a limit exists, then its limit must be unique

If the limit exists, then in a certain decentered neighborhood of X0, the function is bounded

If the limit exists and A>0 (or A<0), then there is a certain decentered neighborhood of X0, and f(x)>0 (or A<0) exists in this decentered neighborhood.

Also known as the reduction theorem and Heine's theorem, it is commonly understood that finding the limit of a function can be transformed into finding the limit of a sequence, and vice versa (here the exponential sequence is included in the function)

When writing infinitesimal quantities, pay attention to the changing trend of its independent variables.

Infinitely small is a variable with 0 as the limit. It is a function, not a number.

0 is the only constant that can be used as an infinitesimal quantity

The definition of infinitesimals also applies to the sequence of numbers

The necessary and sufficient condition for the function f(x) to take A as the limit is: f(x) can be expressed as the sum of A and an infinitesimal α

Use four arithmetic operations

direct substitution method

rationalize the numerator or denominator

Find the highest times

Utilize two important limit formulas

Equivalent to infinitesimal

approximation

common points

If limα(x)/β(x)=0, then α(x) is called the higher-order infinitesimal of β(x), recorded as α(x)=o(β(x))

If limα(x)/β(x)=infinite, then α(x) is called the lower-order infinitesimal of β(x), and is recorded as β(x)=o(α(x))

If limα(x)/β(x)=A, then α(x) is said to be infinitesimal of the same order as β(x), recorded as β(x)=O(α(x)) [it is equivalent when it is 1 infinitesimal

If in a certain limit process, α(x) is the same-order infinitesimal quantity of βk-th power (x) (k>0), then α(x) is said to be the k-order infinitesimal quantity of β(x).

The limit value at this point is equal to its function value

The left limit and the right limit at this point are equal

Discontinuity points can be removed (the limit value is not equal to the function value)

Jump discontinuity point (the left and right limits are not equal)

Same point

no left or right limits

Assume that the functions f(x) and g(x) are both continuous at x0

The direct function is continuous, and the inverse function is also continuous.

The basic elementary function is continuous, and the composite function formed by its composition is also continuous.

The basic elementary functions of its components are all continuous, so when finding the limit of an elementary function at a point within its definition interval, you only need the corresponding function value.

If a function is continuous on a closed interval, then it obtains the maximum and minimum values ​​on the closed interval

A continuous function on a closed interval must be a bounded function on the interval

It is continuous on the closed interval [a,b], and f(a) and f(b) have different signs, then there is at least one point ξ(a<ξ<b) in the open interval (a,b), so that f( ξ)=0

Assume that the function f(x) is continuous on the closed interval [a,b], then on [a,b] it can take any value between its maximum value M and its minimum value m