# A Guide to Real Variables

Steven G. Krantz
Volume: 38
Edition: 1
Pages: 164
https://www.jstor.org/stable/10.4169/j.ctt6wpwwb

1. Front Matter
(pp. I-VIII)
(pp. IX-XIV)
3. Preface
(pp. XV-XVI)
Steven G. Krantz
4. CHAPTER 1 Basics
(pp. 1-12)

Set theory is the bedrock of all of modern mathematics. Asetis a collection of objects. We usually denote a set by an upper case roman letter. IfSis a set andsis one of the objects in that set then we say that s is anelement of Sand we write$s \in S$. Iftis not an element ofSthen we write$t \notin S$.

Some of the sets that we study will be specified just by listing their elements:$S = \{ 2,4,6,8\}$. More often we shall useset-builder notation:$S = \{ x \in \mathbb{R}:4 < {x^2} - 3 < 9\}$. This last is read “the set ofx...

5. CHAPTER 2 Sequences
(pp. 13-22)

Informally, a sequence is an ordered list of numbers:

${a_1},{a_2},{a_3},...$

In more formal treatments, we say that asequenceon a setSis a functionffrom$\mathbb{N}$toS, and we identify$f(j)$with${a_j}$. Although the${a_j}$(or$\{ {a_j}\} _{j = 1}^\infty$) notation is the most common, it is often useful to think of a sequence as a function.

EXAMPLE 2.1.1. Let

$f(j) = 1/{j^2}$.

This function defines the sequence

$\frac{1} {{{1^2}}},\frac{1} {{{2^2}}},\frac{1} {{{3^2}}},...$.

We also write

${a_j} = \frac{1} {{{j^2}}}$.

The primary property of a sequence is its convergence or its non-convergence. We say that a sequence$\{ {a_j}\}$convergesto a numerical limitlif, for every...

6. CHAPTER 3 Series
(pp. 23-40)

Aseriesis, informally speaking, an infinite sum. We write a series as

$\sum\limits_{j = 1}^\infty {{c_j}}$.

We think of the series as meaning

$\sum\limits_{j = 1}^\infty {{c_j}} = {c_1} + {c_2} - {c_3} + ...$.

The basic question about a series is “Does the series converge?” That is to say, does the infinite sum have any meaning? Does it represent some finite real number?

Example 3.1.1. Consider the series

$\sum\limits_{j = 1}^\infty {\frac{1} {{{3^j}}}}$.

Although we do not yet know the rigorous ideas connected with series, we may think about this series heuristically. We may consider the “sum” of this series by adding together finitely many of its terms:

${S_N} = \sum\limits_{j - 1}^N {\frac{1} {{{3^j}}}}$.

It is easy to calculate that${S_N} = \frac{1} {2}(1 - {3^{ - N}})$....

7. CHAPTER 4 The Topology of the Real Line
(pp. 41-54)

Anopen intervalin$\mathbb{R}$is any set of the form

$(a,b) = \{ x \in a < x < b\}$.

Aclosed intervalin$\mathbb{R}$is any set of the form

$[a,b] = \{ x \in \mathbb{R}:a \leqslant x \leqslant b\}$.

The intersection of any two open intervals is either empty (i.e., has no points in it) or is another open interval. The union of two open intervals is either another open interval (if two components intervals overlap) or else is just two disjoint open intervals.

The key property of an open intervals is this:

IfIis an open interval and$x \in I$then there is an$\in > 0$such that

$(x - \in ,x + \in ) \subseteq I$.

Thus any point in an open interval...

8. CHAPTER 5 Limits and the Continuity of Functions
(pp. 55-70)

Let$E \subseteq \mathbb{R}$be a set and letfbe a real-valued function with domainE. Fix a point$P \subseteq \mathbb{R}$that is either inEor is an accumulation point ofE. We say thatfhaslimit$\ell$atP, and we write

$\mathop {lim}\limits_{E \mathrel\backepsilon x \to P} f(x) = \ell$,

with$\ell$a real number, if for each$\in > 0$there is a$\zeta > 0$such that when$x \in E$and$0 < \left| {x - P} \right| < \delta$then

$\left| {f(x) - \ell } \right| < \in$

Example 5.1.2. Let$E = \mathbb{R}\backslash \{ 0\}$and

$f(x) = x.\sin (1/x)$if$x \in E$.

Than${\lim _{x \to 0}}f(x) = 0$. To see this, let$\in > 0$. Choose$\delta = \in$. If$0 < \left| {x - 0} \right| < \delta$then

$\left| {f(x) - 0} \right| = \left| {x.\sin (1/x)} \right|\underline < \left| x \right| < \delta = \in$

as desired. Thus the limit exists and equals 0.

Example 5.1.3. Let$E = \mathbb{R}$and...

9. CHAPTER 6 The Derivative
(pp. 71-84)

Letfbe a function with domain an open interval$x \in I$then the quantity

$\frac{{f(t) - f(x)}} {{t - x}}$

measures the slope of the chord of the graph offthat connects the points$(x,f(x))$and$(t,f(t))$See Figure 6.1. If we let$t \to x$then the limit of the quantity represented by this “Newton quotient” should represent the slope of the graphat the point x. These considerations motivate the definition of the derivative:

Definition 6.1.1. Iffis function with domain an open intervalIand if$x \in I$then the limit

$\mathop {\lim }\limits_{t \to x} \frac{{f(t) - f(x)}} {{t - x}}$,

when it exists, is called thederivativeoffatx....

10. CHAPTER 7 The Integral
(pp. 85-102)

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

Definition 7.1.1. Let$[a,b]$be a closed interval in$\mathbb{R}$. A finite, ordered set of points$p = \{ {x_0},{x_1},{x_2},...,{x_{k - 1}},{x_k}\}$such that

$a = {x_0}\underline < {x_1}\underline < {x_2}...\underline < {x_{k - 1}}\underline < {x_k} = b$

is called apartitionof$[a,b]$. Refer to Figure 7.1.

If$P$is a partition of$[a,b]$, then we let${I_j}$denote the interval$[{x_{j - 1}},{x_j}],j = 1,2,...,k$. The symbol${\Delta _j}$denote thelengthof${I_j}$. Themeshof$P$, denoted by$m(p)$, is defined to be max${\max _j}{\Delta _j}$.

The points of a partition need not be equally spaced, nor must they...

11. CHAPTER 8 Sequences and Series of Functions
(pp. 103-114)

Asequence of functionsis usually written

${f_1}(x),{f_2}(x),...$or$\{ {f_j}(x)\} _{j = 1}^\infty$or$\{ {f_j}\}$.

We will generally assume that the functions${f_j}$all have the same domainS.

Definition 8.1.1. A sequence of functions$\{ {f_j}\} _{j = 1}^\infty$with domain$S \subseteq \mathbb{R}$is said toconverge pointwiseto a limit functionfonSif, for each$x \in S$, the sequence of numbers$\{ {f_j}(x)\}$converges to$f(x)$. We write${\lim _{j \to \infty }}{f_j}(x) = f(x)$.

Example 8.1.2. Define${f_j}(x) \to {x^j}$with domain$S = \{ x:0\underline < x\underline < 1\}$. If$0\underline < x\underline < 1$then${f_j}(x) \to 0$. However,${f_j}(1) \to 1$. Therefore the sequence${f_j}$converges to the fonction

$f(x) - \left\{ \begin{gathered} 0{\text{ if 0}}\underline < x < 1 \hfill \\ 1{\text{ if }}x = 1 \hfill \\ \end{gathered} \right.$

See Figure 8.1.

Here are some of the basic questions that we must ask about...

(pp. 115-128)

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

This section formalizes a general context in which we may do analysis any time we have a reasonable notion of calculating distance. Such a structure will be called a metric:

Definition 9.1.1. Ametric spaceis a pair$(X,\rho )$, whereXis a set and

$\rho :X \times X \to \{ t \in \mathbb{R}:t\underline > 0\}$

is a function satisfying

1.$\forall x,y \in X,\rho (x,y) = \rho (y,x);$

2.$\rho (x,y) = 0$if and only if...

13. Glossary of Terms from Real Variable Theory
(pp. 129-140)
14. Bibliography
(pp. 141-142)
15. Index
(pp. 143-146)
16. Back Matter
(pp. 147-147)