Supremum
Suppose 
is bounded above. A number 
is the supremum of if is an upper bound of and any number less the is not an upper bound of . We write = sup
is bounded above. A number is the supremum of if is an upper bound of and any number less the is not an upper bound of . We write = sup Infimum
Suppose is bounded below. A number 
is the infimum of if is a lower bound of and any number greater the is not a lower bound of . We write = inf
is bounded below. A number is the infimum of if is a lower bound of and any number greater the is not a lower bound of . We write = inf Converges to a number L
A sequence {
} converges to a number 
if
Z 
. The number 
is called the limit of the sequence and the notation 
} converges to a number if Z . The number is called the limit of the sequence and the notation Bounded Below
The set is bounded below if there is a number 
such that 
. The number is called a lower bound of .
is bounded below if there is a number such that . The number is called a lower bound of .Bounded Above
The set is bounded above if there is a number 
such that 
. The number is called an upper bound of .
is bounded above if there is a number such that . The number is called an upper bound of .Bounded
The set is bounded if there is a number such that 
. The number is called a bound for .
is bounded if there is a number such that . The number is called a bound for .
such that 
and 
Interval
A set of real numbers is an interval iff contains at least 2 points and for any two points 
, every real number between 
and 
belong to as well.
of real numbers is an interval iff contains at least 2 points and for any two points , every real number between and belong to as well.Finite
A set 
is finite if it is empty or if its elements can be put in a one to one correspondence with the set {1,2,...,n} for some n.
is finite if it is empty or if its elements can be put in a one to one correspondence with the set {1,2,...,n} for some n.Countably Infinite
The set is countably infinite if its elements can be put in a one to one correspondence with the set of positive integers
is countably infinite if its elements can be put in a one to one correspondence with the set of positive integers
Subsequence
Let 
be a squence and let 
be a strictly increasing sequence of positive integers.
The sequence
is called a subsequence of
be a squence and let be a strictly increasing sequence of positive integers.The sequence
is called a subsequence of 
Order
<, on a set is a relation with the following properties;
Trichotomy, if
, then only 1 of the following holds; 
or 
Transitivity,
, if 
and 
is a relation with the following properties;Trichotomy, if
, then only 1 of the following holds; or Transitivity,
, if and Field
A non empty set, 
, with 
, with 2 operations, + and * w the following properties;
, with , with 2 operations, + and * w the following properties;| Closure |  |  |
| Commutative |  |  |
| Associative |  |  |
| Identity |  |  |
| Inverse |  |  |
| Distributive |  |
that is bounded above has a supremum
is an interval if it has the properties that if 
, and 
, then if 
, and 
Squeeze Theorem for Functions
Let I be an open interval containing point c and suppose 
are functions defined on I except pssibly at c.
Suppose
\{c}
if
then f has limit at c and 
are functions defined on I except pssibly at c.Suppose
\{c}if
then f has limit at c and Subsequential Limit
The Subsequential limit of is any real number, x, such that there exists a subsequence of that converges to x
is any real number, x, such that there exists a subsequence of that converges to xDifferentiable at c
Let 
be an interval, let 
, and let 
. The function 
is differentiable at c provided that the limit
exists. The derivative of at c is the value of the limit and noted by 
be an interval, let , and let . The function is differentiable at c provided that the limitexists. The derivative of
at c is the value of the limit and noted by 
is continuous on 
c,d 
Uniformly continuous
Let be an interval. A function 
is uniformly continuous on if 
that satisfy 
.
be an interval. A function is uniformly continuous on if that satisfy .
Chain Rule
Let be an interval, 
,
is defined on 
that contains 
if
is differentiable on , and is differentiable on then 
is differentiable on and
be an interval, , is defined on that contains if
is differentiable on , and is differentiable on then is differentiable on andIntermediate Value Property
A function defined on an interval has the intermediate value property on if it satisfies the following condition: if 
and 
are distinct points in and 
is any number between 
and 
, then there exists a point 
between and such that 
defined on an interval has the intermediate value property on if it satisfies the following condition: if and are distinct points in and is any number between and , then there exists a point between and such that Limit Superior
if 
is bounded above then the limit superior is defined by 
if not bounded above then the limit superior=
is bounded above then the limit superior is defined by if not bounded above then the limit superior=
Limit Inferior
if is bounded below then the limit inferior is defined by 
if not bounded below then the limit inferior = -
is bounded below then the limit inferior is defined by if not bounded below then the limit inferior = -
Right Hand Limit
is an open interval containing or is at endpoint. is a function defined on except maybe at if
or if is left endpoint, then has right hand limit L at c if 
Left Hand Limit
is an open interval containing or is at endpoint. is a function defined on except maybe at if
or if is right endpoint, then has left hand limit L at c if 
at c
that satisfy 
{
Jump Discontinuity
has jump discontinuity at if the one sided limits exist but are not equal
. 
Removable Discontinuity
at c if 
exists, but 
is either not defined or has a different value from the limit as x approaches c
exists, but is either not defined or has a different value from the limit as x approaches cTagged Partition
A tagged partition 
of an interval 
consists of a partition
{ 
} of along with the set
{
} of points, known as tags, that satisfy 
for 
.
We will express a tagged partition of by
{ 
}
of an interval consists of a partition { } of along with the set{
} of points, known as tags, that satisfy for .We will express a tagged partition
of by { }Norm of Partition, ||P||
The Norm of the partition P, ||P|| = max
If
and 
are partitions of and 
is a refinement of
If
and are partitions of and is a refinement of Riemann Sum
Let 
and let 
be a tagged partition of . The Riemann sum 
of associated with is defined by:
and let be a tagged partition of . The Riemann sum of associated with is defined by:
Riemann Integrable
A function 
is Riemann integrable on if there exists a number with the following property: for each
, such that 
, for all tagged partitions of 
that satisfy 
is Riemann integrable on if there exists a number with the following property: for each , such that , for all tagged partitions of that satisfy Riemann integral
The number L is called the Riemann integral of on and is denoted by the symbol 
, or simply 
.
on and is denoted by the symbol , or simply .
Kartensatzinfo:
Autor: Squiggleart
Oberthema: Mathematics
Thema: Advanced Calculus
Schule / Uni: West Chester University
Veröffentlicht: 25.05.2011
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