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Function, Limit and Continuity of Advanced Mathematics (1)
Definition: 1. If for each number x∈D, the variable y always has a definite y corresponding to it according to certain rules, it is said that x is a function of y, and it is denoted as y=f(x). X is often called an independent variable, Y is called a dependent variable and D is called a domain.

① Symbolic function

② Integer function

Represents the largest integer not exceeding x, such as [3.2]=3, and its basic inequality.

③ Dirichlet function

Definition 2. Let the domain of y=f(u) be D f, the domain of u=g(x) be D g, and the domain of value be R g. If D f∩R g≦? The function y=f[g(x)] is called the coincidence function of functions y=f(u) and u=g(x), and its domain is {x|x∈Dg, g(x)∈D f}

Definition 3. Let the domain of function y=f(x) be d and the domain of value be Ry. If there is a unique x∈D for any y=Ry, so that y=f(x), it is denoted as x=f-1 (y), which is called the inverse function of y=f(x).

Definition 4. Power function (y=xμ), exponential function (y=ax), logarithmic function (y=logax), trigonometric function (y=sinx, y=cosx, y=tanx, y=cotx), inverse trigonometric function (y=arcsinx, y=arccosx, y =

If for any two points on the interval I X 1

Common odd functions:

If t > exists; 0, for any x, there is always f(x+T)=f(x), then y=f(x) is called a periodic function, and the smallest positive number t that keeps the above formula is called the smallest positive period.

If there is M>0, so that for any x∈X, there is always |f(x)|≤M, then f(x) is called a bounded function on X.

? ε& gt; 0,? N>0, when n> is n, there is always | x n-a | < ε.

? ε& gt; 0,? X>0, when x> is at X, there is always | f (x)-a | < ε.

? ε& gt; 0,? X>0, when x

? ε& gt; 0,? X>0, when |x| > is at X, there is always | f (x)-a | < ε.

? ε& gt; 0,? δ& gt; 0, when 0 < | x-x0 | < At δ, there is always | f (x)-a | < ε.

If n exists: when n> when n, x n ≤y n ≤z n, and

rule

Monotone bounded sequence must have a limit.

Then α(x) is called the high-order infinitesimal of β(x), which is denoted as α(x)=o[β(x)].

Then α(x) is called the low-order infinitesimal of β(x)

Then α(x) is called the same order infinitesimal of β(x)

Then α(x) is called equal-order infinitesimal of β(x), and it is denoted as α(x)~β(x).

In particular, if

Then α(x) is called the k-order infinitesimal of β(x)

In the same limit process, if f(x) is infinite, then 1/f(x) is infinite. On the other hand, if f(x) is infinitesimal and f(x)≠0, then 1/f(x) is infinite.

If limα(x)=0, limβ(x)=∞, and limα(x)β(x)=A, then lim α (x) β (x) = e a.

And lim α1(x)/β (x) = a ≦-1,then α (x)+β (x) ~ α1(x)+β1(x).

If LIMf(x) = LIMF (x) = 0 (∞), and f(x) and g(x) are derivable in the centripetal field of x0, and g'(x)≠0, limf'(x)/g'(x) exists (or is infinite), then

Where r n (x) = o (x-x 0) n 0) n.

Commonly used inequalities:

Define 1, if

Then it is said that y=f(x) is continuous at point x 0.

If f(x) is defined in the centripetal field of x 0, but it is discontinuous at x 0, then x 0 is called the discontinuous point of f(x).