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
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4A
Continuous Functions
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
x a
real number $ % ! such that fo
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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
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Limits of Sequences
The Limit in and Its Basic Properties
We first discuss sequences in .
2.1 Definition
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
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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.
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The Completeness Property
1.1 Definition Let W be a set.
(a) A number ? is said to
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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
_
8"
_
write ! 08 ! 08 x. Given an infinite ser
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Infinite Series
Convergence of Infinite Series
Let B8 be a sequence in . We call ! B8 the infinite series generalized by
_
8"
B8 . Given an infinite series ! B8 , we call for each 5 , the finite sum =5 ! B8
_
5
8"
8"
to be the 5 th partial sum. Then
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Improper and Infinite Integrals
Improper 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
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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 :
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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
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Differentiation in
Derivatives
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 % !,