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The subject of real analysis dates to the mid-nineteenth century - the days of Riemann and Cauchy and Weierstrass. Real analysis grew up as a way to make the calculus rigorous. Today the two subjects are intertwined in most people's minds. Yet calculus is only the first step of a long journey, and real analysis is one of the first great triumphs along that road. In real analysis we learn the rigorous theories of sequences and series, and the profound new insights that these tools make possible. We learn of the completeness of the real number system, and how this property makes the real numbers the natural set of limit points for the rational numbers. We learn of compact sets and uniform convergence. The great classical examples, such as the Weierstrass nowhere-differentiable function and the Cantor set, are part of the bedrock of the subject. Of course complete and rigorous treatments of the derivative and the integral are essential parts of this process. The Weierstrass approximation theorem, the Riemann integral, the Cauchy property for sequences, and many other deep ideas round out the picture of a powerful set of tools.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Basics

Abstract
Set theory is the bedrock of all of modern mathematics. A set is a collection of objects. We usually denote a set by an upper case roman letter. If S is a set and s is one of the objects in that set, then we say that s is an element of S and we write sS. If Pt is not an element of S, then we write tS.
Steven G. Krantz

Chapter 2. Sequences

Abstract
Informally, a sequence is a list of numbers:
$$ {a_1},{a_2},{a_3}....$$
In more formal treatments, we say that a sequence on a set S is a function f from ℕ to S, and we identify f(j) with a j . Such precision will not be required here.
Steven G. Krantz

Chapter 3. Series

Abstract
A series is, informally speaking, an infinite sum. We write a series as
$$ \sum\limits_{j = 1}^\infty {{c_j}.}$$
Steven G. Krantz

Chapter 4. The Topology of the Real Line

Without Abstract
Steven G. Krantz

Chapter 5. Limits and the Continuity of Functions

Abstract
Definition 5.1 Let E ℝ be a set and let f be a real-valued function with domain E. Fix a point P∈ ℝ that is either in E or is an accumulation point of E. We say that f has limit l at P, and we write
$$ \mathop {\lim }\limits_{E \mathrel\backepsilon x \to P} f(x) = \ell ,$$
with l a real number, if for each ∈ > 0 there is a δ > 0 such that when xE and 0 <-P< δ then
$$ |f(x) - \ell | < \in .$$
Steven G. Krantz

Chapter 6. The Derivative

Abstract
Let f be a function with domain an open interval I. If xI, then the quantity
$$ \frac{{f(t) - f(x)}}{{t - x}}$$
measures the slope of the chord of the graph of f that connects the points (x, f(x)) and (t, f(t)). If we let tx, then the limit of the quantity represented by this “Newton quotient” should represent the slope of the graph at the point x.
Steven G. Krantz

Chapter 7. The Integral

Abstract
The integral is a generalization of the summation process. That is the point of view that we shall take in this chapter.
Steven G. Krantz

Chapter 8. Sequences and Series of Functions

Abstract
A sequence of functions is usually written
$$ {f_1}(x),{f_2}(x), \ldots or \{ {f_j}\} _{j = 1}^\infty .$$
We will generally assume that the functions f j all have the same domain S.
Steven G. Krantz

Chapter 9. Some Special Functions

Abstract
A series of the form
$$ \sum\limits_{j = 0}^\infty {{a_j}{{(x - c)}^j}}$$
is called a power series expanded about the point c. Our first task is to determine the nature of the set on which a power series converges.
Steven G. Krantz

Chapter 10. Advanced Topics

Abstract
Part of the power of modern analysis is to look at things from an abstract point of view. This provides both unity and clarity, and also treats all dimensions at once. We shall endeavor to make these points clear as we proceed.
Steven G. Krantz

Chapter 11. Differential Equations

Abstract
Differential equations are the heart and soul of analysis. Virtually any law of physics, engineering, biology, or chemistry can be expressed as a differential equation — and frequently as a first-order equation (i.e., an equation involving only first derivatives). Much of mathematical analysis has been developed in order to find techniques for solving differential equations.
Steven G. Krantz

Backmatter

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