Classification of real numbers

The size relationship between real numbers

n-digit deficiency approximation and excess approximation of x

Six properties of real numbers

Definition of absolute value of real number a

Absolute value properties of real numbers

Classification of intervals

Classification of Neighborhoods

Upper bound (lower bound) definition

Definition of upper (lower) boundary

Bounded set definition

Unbounded set definition

principle of certainty

Abnormal boundaries

Promotional Determination Principle

In a process of change, the quantity that changes is called a variable (in mathematics, the variable is x, and y changes with the change of the value of x). Some values ​​do not change with the variable, and they are called constants.

We often use y=f(x), x∈D to represent a function. Therefore, we say that two functions are the same <=> and they have the same domain and corresponding rules.

The domain of a function is usually the set of independent variable values ​​that make the operation meaningful. It is usually called the existential domain. It can simply be said that "function y=f(x) or function f"

The domain of a function is usually the set of independent variable values ​​that make the operation meaningful. It is usually called the existential domain. It can simply be said that "function y=f(x) or function f"

The domain of a function is usually the set of independent variable values ​​that make the operation meaningful. It is usually called the existential domain. It can simply be said that "function y=f(x) or function f"

The function f gives a single-valued correspondence between the point set D on the x-axis and the point set M on the y-axis, also called a mapping. For a∈D, f(a) is called the image of a under the mapping f, and a is called the original image of f(a).

In the function definition, for each x∈D, there can only be a unique y value corresponding to it. The function defined in this way is called a single-valued function. If the same x value can correspond to more than one y value, it is called This function is a multi-valued function.

Replenish

symbolic function

Dirichlet function

Riemann function

Given two functions f, x∈D₁ and g, x∈D₂. Note D=D₁∩D₂, and assume D≠∅, we define the four arithmetic operations of f and g on D as follows

F(x)=f(x) g(x),x∈D

G(x)=f(x)-g(x),x∈D

H(x)=f(x)g(x),x∈D

L(x)=f(x)/g(x),x∈D*

Note: If D=D₁∩D₂=∅, f and g cannot perform four arithmetic operations

y=f(g(x)), x∈E* is called the composite function of functions f and g, and is called the outer function of f, the inner function of g, and u is the intermediate variable.

Composite functions can also be composed of multiple functions.

Note: If and only if E*≠∅ (i.e. D∩g(E)≠∅), functions f and g can be composited

Suppose the function y=f(x), x∈D satisfies: for every y in the value range f(D), there is only one x in D, so that f(x)=y, then according to this corresponding rule, we get A function defined on f(D) is called the inverse function of f, denoted as f -1: f(D)→D(yl→x)

Note 1: Function f has an inverse function, which means that f is a one-to-one mapping between D and f(D). We call f-1 the inverse mapping of mapping f.

Note 2: y=f-1(x),x∈f(D)

Constant function y=c (c is a constant)

Power function y=x raised to the power α (α is a real number)

Exponential function y=a raised to the power of x (a>0, a≠1)

Logarithmic function y=logₐx(a>0,a≠1)

Trigonometric functions y=sinx (sine function) y=cosx (cosine function) y=tanx (tangent function) y=cotx (cotangent function)

Inverse trigonometric function y=arcsinx (inverse sine function) y=arccosx (inverse cosine function) y=arctanx (arctangent function) y=arccotx (inverse cotangent function)

Six categories of basic elementary functions

Functions obtained from basic elementary functions through a finite number of four arithmetic operations and compound operations are collectively called elementary functions. Functions that are not elementary functions are called non-elementary functions.

Given a real number a>0, a≠1, assuming x is an irrational number, we stipulate that a raised to the power of x =

Definition 1 Let f be a function defined on D. If there is a number M(L) such that for every x∈D, f(x)≤M(f(x)≥L), then f is called a function on D. has an upper (lower) bound function, M(L) is called an upper (lower) bound f on D

Definition 2 Let f be a function defined on D. If there is a positive number M such that for every x∈D, │f(x)│≤M, then f is called a bounded function on D.

(i)inff(x)(x∈D) infg(x)(x∈D)≤inf│f(x) g(x)│(x∈D)

(ii) sup│f(x) g(x)│(x∈D)≤supf(x)(x∈D) supg(x)(x∈D)

Suppose y=f(x), x∈D is a strictly increasing (decreasing) function, then f must have an inverse function f-1, and f-1 is also a strictly increasing (decreasing) function in the domain f(D)

Let D be a number set symmetrical to the origin, and f be a function defined on D. If for each x∈D, f(-x)=-f(x), then f is called odd (even) on D. function

Let f be a function defined on the number set D. If there is σ>0, so that for all x∈D, xσ∈D, f(x±σ)=f(x), then f is called a periodic function. σ is called a period of f