# Introduction to Analysis

• I. N. Bronshtein
• K. A. Semendyayev

## Abstract

Rational numbers. All integral and fractional numbers (positive, negative and zero) are called rational. Rational numbers form an infinite set with the following properties:
1. (1)

This set is ordered, i.e., for every two rational numbers it is possible to indicate which of them is greater.

2. (2)

This set is everywhere dense, i.e., between every two rational numbers a and b (a < b) there exists at least one rational number c (a < c < b); hence there exists also an infinite set of rational numbers between a and b.

3. (3)

The arithmetic operations (addition, subtraction, multiplication and division) with two arbitrary rational numbers are always performable and give as the result a rational number. Division by zero is an exception; it is not allowed. The expression a/0 has no definite meaning, for there exists no definite number b satisfying the equation b·0 = a (if a = 0, then b can be arbitrary, and if a = 0, then b does not exist)(1).

4. (4)

Every rational number can be represented by a decimal fraction (which may be finite or recurring).

## Keywords

Rational Number Connected Domain Trigonometric Function Algebraic Function Convergent Series
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

1. (1).
The equality a/0 = ∞ (infinity) which is often used does not mean that this division is performable (infinity is not a number); it is simply an abbreviation for the statement: “if the divisor tends to zero, then the absolute value of the quotient increases beyond bound”.Google Scholar
2. (1).
We consider here only infinite sequences.Google Scholar
3. (1).
The numbers an may be equal to A for certain values of n.Google Scholar
4. (1).
A sequence satisfying this condition is said to be a fundamental sequence.Google Scholar
5. (2).
We consider here only functions of a real variable. For functions of a complex variable see pp. 590–605.Google Scholar
6. (1).
The arrows mean that the end point in the arrowhead does not belong to the graph.Google Scholar
7. (1).
The tables of elementary functions are given on pp. 19–83 and their graphs on pp. 96–116.Google Scholar
8. (1).
An argument x of a trigonometric function sin x, cos x, tan x, … in mathematical analysis is understood to be an arbitrary number and not an angle or an arc of a circle (as in elementary trigonometry). Trigonometric functions can be defined purely analytically, without geometric notions; for example, sin x may be defined by its expansion into a power series (see p. 388) or as a solution of the differential equation d2y/dx2 + y = 0 with the initial condition: x = 0, y = 0, dy/dx = 1. In this sense, the argument of a trigonometric function is equal numerically to its circular measure expressed in radians. Thus to evaluate trigonometric functions, trigonometric tables can be used, but the radian measure of angle should be changed into degree.Google Scholar
9. (1).
Sometimes one function can be defined in many ways.Google Scholar
10. (1).
The monotone functions defined as above are sometimes called monotone in the wider sense. A function satisfying the condition f(x1) < f(x2) or f(x1) > f(x2) (a sharp inequality), for every x1 < x2, is said to be strictly monotone. The function in Fig. 268a is strictly monotone increasing and the function in Fig. 268b is monotone in the wider sense (it is constant in the interval AB).Google Scholar
11. (2).
The notion of a limit can also be introduced for functions defined in more complicated domains.Google Scholar
12. (1).
13. (2).
If a is a boundary point of the considered domain, then this double inequality should be replaced by a simple one: a − η < x or x < a + η.Google Scholar
14. (1).
At the point a the functions φ(x) and ψ(x) need not be defined.Google Scholar
15. (1).
The limit of a constant function is equal to this constant.Google Scholar
16. (1).
An arrow in the figure denotes that the point in the arrowhead does not belong to the graph; a heavy point denotes a point belonging to the graph.Google Scholar
17. (2).
The second condition can be replaced by the following equivalent one: for any infinitesimal α, the difference β= f(a + α)—f(a) is an infinitesimal, i.e., infinitesimal increments of the argument induce infinitesimal increments of the function.Google Scholar
18. (3).
If a function is defined only on one side of an argument x=a (as, for example, + √x for x = 0 or arc cos x for x = 1), then it is said to break off.Google Scholar
19. (1).
For the notation f(a − 0), f(a + 0), see p. 329.Google Scholar
20. (1).
21. (1).
The concept of oscillation of a function can be extended to functions without a greatest and least value.Google Scholar
22. (1).
Such a system can be regarded as a current point of an n-dimensional space.Google Scholar
23. (2).
Functions of three or more variables cannot be represented geometrically. But, by analogy with three dimensional space, we introduce also the concept of a hypersurface in the n-dimensional space.Google Scholar
24. (1).
Fig. 277 shows some simplest examples of connected domains of two variables with their names. The domains are lined; if a boundary belongs to the domain, it is drawn by a continuous line; if a boundary does not belong to the domain, it is drawn by a dotted line.Google Scholar
25. (1).
For the rank of a matrix, see p. 176.Google Scholar
26. (1).
Only functions defined in a connected domain are considered here.Google Scholar
27. (1).
28. (1).
If two series a1 + a2 + … + an + … and b1 + b2 + … + bn + … are convergent and at least one of them is absolutely convergent, then the series obtained from multiplication is convergent though not necessarily absolutely convergent.Google Scholar
29. (1).
From a formula for the remainder of a Maclaurin’s series, see p. 386.Google Scholar
30. (1).
If these limits do not exist, then lim sup should be taken.Google Scholar