[advanced mathematics] [1] function, limit, continuity

1. Mapping and function

1.1 mapping

1.2 Composite mapping

There are two mappings

1.3 function

1.3.1 Absolute function

1.3.2 Symbolic function

1.3.3 Integral function

1.3.4 Piecewise functions

A function should be expressed by several formulas , This is in different variation ranges of independent variables , The corresponding rule is a function expressed by different formulas , It is usually called piecewise function

1.4 Characteristics of functions

Notice that there is equal to

1.5 Parity function

• A function that defines domain symmetry , It can be divided into an odd function and an even function

1.6 Elementary function

1.7 Anti hyperbolic function

1.8 Power exponential function

2. The limit of the sequence

2.1 Definition of sequence limit

2.2 Properties of convergent sequence

• The uniqueness of the limit , If the sequence { $x_n$} convergence , So its limit is unique
• Boundedness of convergent sequence , If the sequence { $x_n$} convergence , Then the sequence must be bounded
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3. The limit of a function

3.1 Definition of function limit 【 When the independent variable tends to a finite value 】

3.2 Unilateral limit

3.3 Definition of function limit 【 When the independent variable tends to infinity 】

3.4 Properties of function limit

4. Infinitesimal and infinitesimal

4.1 Infinitesimal definition

4.2 Definition of infinity

4.3 Theorem

5. Limit algorithm

• The sum of two infinitesimals is infinitesimal , The sum of finite infinitesimals is also infinitesimal
• The product of bounded function and infinitesimal is infinitesimal
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  Be careful , if $y_n$=0, Then the algorithm about the limit of quotient cannot be applied
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6. Limit existence criteria Two important limits

6.1 Two important limits

$$\underset{x\to 0}{\lim }\frac{sinx}{x}=1$$
$$\underset{x\to \infty }{\lim }(1+\frac{1}{x}{)}^x=e$$

6.2 Limit existence criteria

6.2.1 Criteria for the existence of sequence limits 1

6.2.2 The existence criterion of function limit 1【 Pinch rule 】

6.2.3 Criteria for the existence of sequence limits 2

Monotone bounded sequence has limit
Be careful : Bounded sequences do not necessarily have limits

6.2.4 The existence criterion of function limit 2【 The existence criterion of Cauchy limit ( Cauchy convergence principle )】

7. Infinitesimal comparison

7.1 Definition

7.2 Equivalent infinitesimal theorem

7.3 The addition and subtraction method uses the equivalent infinitesimal

7.4 The equivalent infinitesimal 【 Commonly used 】

7.5 matters needing attention

• Infinitesimal applies to multiplication and division

8. Continuity and discontinuities of functions

8.1 Definition of continuity

Mode one

Mode two

8.2 Break point

8.2.1 Infinite discontinuity 【 The second category 】

8.2.2 Oscillation breakpoint 【 The second category 】

8.2.3 Go to the break point 【 The first category 】

8.2.4 Break point between jumps 【 The first category 】

8.2.5 The first type of discontinuity + The second kind of discontinuity

9. Operation of continuous function and continuity of elementary function

9.1 Continuous function theorem

9.2 Continuity of elementary function

• Basic elementary functions are continuous in their domain

10. The properties of continuous functions on closed intervals

10.1 Boundedness and maximum minimum theorem

10.2 Zero point theorem and intermediate value theorem

10.2.1 Zero point theorem

10.2.2 Intermediate value theorem

10.3 Uniform continuity