Kimberly Zhang Analysis Useful Things |a + b| |a| + |b| | |a| |b| | |a b| Definitions L is an upper bound for S iff L s for all s S. L is the least upper bound for S iff L is an upper bound for S and if M is also an upper bound for S, then L M. A sequence
Local Properties of Continuous Functions
Let 0 be a function with domain H0 : and range V0 ; .
4.1 Definition of Limits of Functions
Let a H0 . We say lim 0 x b ; if for each % !, there exists a
real number $ % ! such that fo
The Topology of :
Open and Closed Sets
Let x : and < !. We denote by F< x y : , mx ym < the ball
centered at x with radius <. Here
mx ym B" C" # B# C# # B: C: # "#
is the Euclidean distance between x and y.
A set Y of : is called a neighborhood of a
Limits of Sequences
The Limit in and Its Basic Properties
We first discuss sequences in .
Let B8 be a sequence in . A number B is a lmit of B8 , written as limB8 B,
if for each % !, there exists a natural number O% ! such that for all
and the Completeness Property
The set of real numbers : rational numbers and irrational numbers. There is
a bijection between and the set of points in the real axis.
The Completeness Property
1.1 Definition Let W be a set.
(a) A number ? is said to
Series of Functions
Convergence of Series of Functions
Let 08 be a sequence of functions defined on a domain H : with value in
. We call ! 08 x the series of functions generalized by 08 . Sometimes we
write ! 08 ! 08 x. Given an infinite ser
Convergence of Infinite Series
Let B8 be a sequence in . We call ! B8 the infinite series generalized by
B8 . Given an infinite series ! B8 , we call for each 5 , the finite sum =5 ! B8
to be the 5 th partial sum. Then
Improper and Infinite Integrals
Let 0 be a function defined on + , and is unbounded near +. If 0 is integrable
on - , for all - with + - , and the limit
lim ( 0 .B
exists, then we say the limit is the improper integral of 0
Riemann Integration in
Definition of the Riemann Integral
Throughout of this chapter, we assume 0 to be bounded functions defined on
N + , We shall define the Riemann integration of 0 over N
A partition T of N is a finite set of real numbers B" B#
Differentiation in :
The Definition of Derivatives
In the following we will assume that 0 is a function from a domain
E : to ; and c is an interior point of E.
6.1 Definition of Directional Derivatives
Let u be a vector in : . A vector Pu ; is said t
In the following, we shall consider scalar function 0 defined on an interval in .
5.1 Definition of Derivatives
Let 0 be defined in + , and let - + , We say a real number P is the
derivative of 0 at - if for every % !,