1
Chapter 1
Functions, Limits, Continuity and Differentiability
1
Functions of a single real variable
(p.157 – p.158, p.173 – p.176, p.193 – p.197)
1.1
Sets
A set is a collection of distinct objects called
elements
or
members
of that set. For example,
{ }
1,2,3,4,5
A
=
is a set and a list of all its elements is given. In general, we use the notation
{
xx
processes certain properties}to denote a set of objects that share some common properties. Also, if
e
is an element of a set
A
, we write
A
e
∈
(read as
e belongs to A
).
Given two sets
A
and
B
, we say that
A
is a
subset
of
E
(denoted by
A
E
⊂
) if all elements of
A
belong to
E
.
Example 1
We use the symbol
R
to denote the set which contains exactly all the real numbers. Also, the following
symbols are frequently used to describe the corresponding subsets of real numbers (
a
,
b
are two distinct
real numbers):
( )
[]
()
,{

}
[,) {

}

}
[, ) {

}

}
ab
x
a x b
x
x
ax
x
a
x
a
=
∈<
<
=
∈≤
<
=
≤
∞= ∈
≥
−∞
=
∈
<
R
R
R
R
R
(The other subsets like
( ) ( )
( , ],
,
,(
, ],
,
ab a
a
∞−
∞ −
∞
∞
are defined similarly). These sets are usually called
intervals
. In our discussion, most of the sets we consider are intervals.
,
1.2
Functions
A function
f
is a rule of correspondence that associates with each object
x
in one set
A
(called the
domain
of
f
) a unique value
f
x
from a second set
B
(called the codomain of
f
). The set of values so obtained
is called the
range
of the function.
f
domain
codomain
range
It is customary to write
f
as:
:
f
AB
→