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## Über dieses Buch

Based on courses given at Eötvös Loránd University (Hungary) over the past 30 years, this introductory textbook develops the central concepts of the analysis of functions of one variable — systematically, with many examples and illustrations, and in a manner that builds upon, and sharpens, the student’s mathematical intuition. The book provides a solid grounding in the basics of logic and proofs, sets, and real numbers, in preparation for a study of the main topics: limits, continuity, rational functions and transcendental functions, differentiation, and integration. Numerous applications to other areas of mathematics, and to physics, are given, thereby demonstrating the practical scope and power of the theoretical concepts treated.

In the spirit of learning-by-doing, Real Analysis includes more than 500 engaging exercises for the student keen on mastering the basics of analysis. The wealth of material, and modular organization, of the book make it adaptable as a textbook for courses of various levels; the hints and solutions provided for the more challenging exercises make it ideal for independent study.

## Inhaltsverzeichnis

### Chapter 1. A Brief Historical Introduction

The first problems belonging properly to mathematical analysis arose during fifth century bce, when Greek mathematicians became interested in the properties of various curved shapes and surfaces. The problem of squaring a circle (that is, constructing a square of the same area as a given circle with only a compass and straightedge) was well known by the second half of the century, and Hippias had already discovered a curve called the quadratix during an attempt at a solution. Hippocrates was also active during the second half of the fifth century bce, and he defined the areas of several regions bound by curves (“Hippocratic lunes”).
Miklós Laczkovich, Vera T. Sós

### Chapter 2. Basic Concepts

In former times, mathematics was defined as the science concerned with numbers and figures. (This is reflected in the title of the classic book by Hans Rademacher and Otto Toeplitz, Von Zahlen und Figuren, literally On Numbers and Figures [6].) Nowadays, however, such a definition will not do, for modern algebra deals with abstract structures instead of numbers, and some branches of geometry study objects that barely resemble any figure in the plane or in space. Other branches of mathematics, including analysis, discrete mathematics, and probability theory, also study objects that we would not call numbers or figures. All we can say about the objects studied in mathematics is that generally, they are abstractions from the real world (but not always).
Miklós Laczkovich, Vera T. Sós

### Chapter 3. Real Numbers

What are the real numbers? The usual answer is that they comprise the rational and irrational numbers. That is correct, but what are the irrational numbers? They are the numbers whose infinite decimal expansions are infinite and nonrepeating. But for this, we need to know precisely what an infinite decimal expansion is.
Miklós Laczkovich, Vera T. Sós

### Chapter 4. Infinite Sequences I

In this chapter, we will be dealing with sequences of real numbers. For brevity, by a sequence we shall mean an infinite sequence whose terms are all real numbers.
Miklós Laczkovich, Vera T. Sós

### Chapter 5. Infinite Sequences II

Finding the limit of a sequence is generally a difficult task. Sometimes, just determining whether a sequence has a limit is tough. Consider the sequence (18) in Example 4.​1, that is, let a n be the nth digit in the decimal expansion of $$\sqrt{2}$$. We know that (a n ) does not have a limit. But does the sequence $$c_{n} = \root{n}\of{a_{n}}$$ have a limit? First of all, let us note that a n  ≥ 1, and thus c n  ≥ 1 for infinitely many n. Now if there are infinitely many zeros among the terms a n , then c n  = 0 also holds for infinitely many n, so the sequence (c n ) is divergent. However, if there are only finitely many zeros among the terms a n , that is, a n ≠ 0 for all n > n 0, then 1 ≤ a n  ≤ 9, and so $$1 \leq c_{n} \leq \root{n}\of{9}$$ also holds if n > n 0. By Theorem 4.17, $$\root{n}\of{9} \rightarrow 1$$. Thus for a given ɛ > 0, there is an n 1 such that $$\root{n}\of{9} <1+\varepsilon$$ for all n > n 1. So if $$n>\max (n_{0},n_{1})$$, then $$1 \leq c_{n} <1+\varepsilon$$, and thus c n  → 1.
Miklós Laczkovich, Vera T. Sós

### Chapter 6. Infinite Sequences III

In Theorem 4.10, we proved that for a sequence to converge, a necessary condition is the boundedness of the sequence, and in our example of the sequence (−1) n , we saw that boundedness is not a sufficient condition for convergence.
Miklós Laczkovich, Vera T. Sós

### Chapter 7. Rudiments of Infinite Series

If we add infinitely many numbers (more precisely, if we take the sum of an infinite sequence of numbers), then we get an infinite series.
Miklós Laczkovich, Vera T. Sós

### Chapter 8. Countable Sets

While we were talking about sequences, we noted that care needs to be taken in distinguishing the sequence (a n ) from the set of its terms {a n }. We will say that the sequence (a n ) lists the elements of H if H = { a n }. (The elements of the set H and thus the terms of (a n ) can be arbitrary; we do not restrict ourselves to sequences of real numbers.)
Miklós Laczkovich, Vera T. Sós

### Chapter 9. Real-Valued Functions of One Real Variable

Consider a function $$f: A \rightarrow B$$. As we stated earlier, by this we mean that for every element a of the set A, there exists a corresponding b ∈ B, which is denoted by b = f(a).
Miklós Laczkovich, Vera T. Sós

### Chapter 10. Continuity and Limits of Functions

If we want to compute the value of a specific function at some point a, it may happen that we can compute only the values of the function near a. Consider, for example, the distance a free-falling object covers. This is given by the equation $$s(t) = g \cdot t^{2}/2$$, where t is the time elapsed, and g is the gravitational constant. Knowing this equation, we can easily compute the value of s(t). If, however, we want to calculate s(t) at a particular time t = a by measuring the time, then we will not be able to calculate the precise distance corresponding to this given time; we will obtain only a better or worse approximation—depending on the precision of our instruments. However, if we are careful, we will hope that if we use the value of t that we get from the measurement to recover s(t), the result will be close to the original s(a). In essence, such difficulties always arise when we are trying to find some data with the help of another measured quantity. At those times, we assume that if our measured quantity differs from the real quantity by a very small amount, then the value computed from it will also be very close to its actual value.
Miklós Laczkovich, Vera T. Sós

### Chapter 11. Various Important Classes of Functions (Elementary Functions)

In this chapter, we will familiarize ourselves with the most commonly occurring functions in mathematics and in applications of mathematics to the sciences. These are the polynomials, rational functions, exponential, power, and logarithm functions, trigonometric functions, hyperbolic functions, and their inverses. We call the functions that we can get from the above functions using basic operations and composition elementary functions.
Miklós Laczkovich, Vera T. Sós

### Chapter 12. Differentiation

Consider a point that is moving on a line, and let s(t) denote the location of the point on the line at time t. Back when we talked about real-life problems that could lead to the definition of limits (see Chapter 9, p. 121), we saw that the definition ofinstantaneous velocity required taking the limit of the fraction $$\big(s(t) - s(t_{0})\big)/(t - t_{0})$$ in t 0.
Miklós Laczkovich, Vera T. Sós

### Chapter 13. Applications of Differentiation

The following theorem gives a useful method for determining critical limits.
Miklós Laczkovich, Vera T. Sós

### Chapter 14. The Definite Integral

In the previous chapter of our book, we became acquainted with the concept of the indefinite integral: the collection of primitive functions of f was called the indefinite integral of f. Now we introduce a very different kind of concept that we also call integrals—definite integrals, to be precise. This concept, in contrast to that of the indefinite integral, assigns numbers to functions (and not a family of functions). In the next chapter, we will see that as the name integral that they share indicates, there is a strong connection between the two concepts of integrals.
Miklós Laczkovich, Vera T. Sós

### Chapter 15. Integration

In this chapter, we will familiarize ourselves with the most important methods for computing integrals, which will also make the link between definite and indefinite integrals clear.
Miklós Laczkovich, Vera T. Sós

### Chapter 16. Applications of Integration

One of the main goals of mathematical analysis, besides applications in physics, is to compute the measure of sets (arc length, area, surface area, and volume). We have already spent time computing arc lengths, but only for graphs of functions. We saw examples of computing the area of certain shapes (mostly regions under graphs), and at the same time, we got a taste of computing volumes when we determined the volume of a sphere (see item 2 in Example 13.23). We also noted, however, that in computing area, some theoretical problems need to be addressed (as mentioned in point 5 of Remark 14.10). In this chapter, we turn to a systematic discussion of these questions.
Miklós Laczkovich, Vera T. Sós

### Chapter 17. Functions of Bounded Variation

We know that if f is integrable, then the lower and upper sums of every partition F approximate its integral from below and above, and so the difference between either sum and the integral is at most $$S_{F} - s_{F} =\varOmega _{F}$$, the oscillatory sum corresponding to F.
Thus the oscillatory sum is an upper bound for the difference between the approximating sums and the integral.
We also know that if f is integrable, then the oscillating sum can become smaller than any fixed positive number for a sufficiently fine partition (see Theorem 14.23).
Miklós Laczkovich, Vera T. Sós

### Chapter 18. The Stieltjes Integral

In this chapter we discuss a generalization of the Riemann integral that is often used in both theoretical and applied mathematics. Stieltjes originally introduced this concept to deal with infinite continued fractions, but it was soon apparent that the concept is useful in other areas of mathematics—and thus in mathematical physics, probability, and number theory, independently of its role in continued fractions. We illustrate the usefulness of the concept with two simple examples.
Miklós Laczkovich, Vera T. Sós

### Chapter 19. The Improper Integral

Until now, we have dealt only with integrals of functions that are defined in some closed and bounded interval (except, perhaps, for finitely many points of the interval) and are bounded on that interval. These restrictions are sometimes too strict; there are problems whose solutions require us to integrate functions on unbounded intervals, or that themselves might not be bounded.

Miklós Laczkovich, Vera T. Sós

### Erratum

Miklós Laczkovich, Vera T. Sós

### Backmatter

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