10-07-2012, 12:42 PM
Advanced Calculus with Applications in Statistics
Advanced Calculus 2nd Ed.pdf (Size: 7.69 MB / Downloads: 194)
SOME BASIC TOPOLOGICAL CONCEPTS
The field of topology is an abstract study that evolved as an independent
discipline in response to certain problems in classical analysis and geometry.
It provides a unifying theory that can be used in many diverse branches of
mathematics. In this section, we present a brief account of some basic
definitions and results in the so-called point-set topology.
EXAMPLES IN PROBABILITY AND STATISTICS
EXAMPLE 1.7.1. In probability theory, events are considered as subsets in
a sample space , which consists of all the possible outcomes of an experiment.
A Borel field of events Žalso called a -field. in is a collection B of
events with the following properties:
Limits and Continuity of Functions
The notions of limits and continuity of functions lie at the kernel of calculus.
The general concept of continuity is very old in mathematics. It had its
inception long ago in ancient Greece. We owe to Aristotle Ž384322 B.C..
the first known definition of continuity: ‘‘A thing is continuous when of any
two successive parts the limits at which they touch are one and the same and
are, as the word implies, held together’’ Žsee Smith, 1958, page 93.. Our
present definitions of limits and continuity of functions, however, are substantially
those given by Augustin-Louis Cauchy Ž17891857..
In this chapter we introduce the concepts of limits and continuity of
real-valued functions, and study some of their properties. The domains of
definition of the functions will be subsets of R, the set of real numbers.
A typical subset of R will be denoted by D.
LIMITS OF A FUNCTION
Before defining the notion of a limit of a function, let us understand what is
meant by the notation x™a, where a and x are elements in R. If a is finite,
then x™a means that x can have values that belong to a neighborhood
NrŽa. of a Žsee Definition 1.6.1. for any r0, but xa, that is, 0 xya
r. Such a neighborhood is called a deleted neighborhood of a, that is, a
neighborhood from which the point a has been removed. If a is infinite Žy
or q., then x™a indicates that x can get larger and larger without any
constraint on the extent of its increase. Thus x can have values greater than
any positive number. In either case, whether a is finite or infinite, we say that
x tends to a or approaches a.