Graphics Reference
In-Depth Information
The ranges of these seven functions exhibit the seven kinds of
interval
on the real line.
Intervals
, or connected sets, on the real line are subsets of the real
line which contain all the real numbers lying between any two points of
the subset. The set
I
is an
interval
if when
r
and
s
I
, and
r
s
, every
x
such that
r
x
s
also belongs to
I
.
The seven types of interval are distinguished by their boundedness
and their boundaries.
Bounded above and below
(i) singleton point
a
(ii) closed interval
[
a
,
b
]
x
a
x
b
x
a
x
b
(iii) open interval
(
a
,
b
)
(iv) half-open interval
[
a
,
b
)
x
a
x
b
(
a
,
b
]
x
a
x
b
Bounded above
or
below
,
but not both
(v) open half-ray
(
a
,
)
x
a
x
(
,
a
)
x
x
a
(vi) closed half-ray
[
a
,
)
x
a
x
(
,
a
]
x
x
a
Unbounded
(vii) thewholereal line
R
Theword
open
as we have used it here indicates that there is space
within
each open set around each point of the set. For example, if
c
b
. This is also described by
saying that an open set contains a neighbourhood of each of its points.
Theword
closed
as we have used it here indicates that every
convergent sequence within each closed set converges to a point of the
set. (See qn 3.78.)
The whole real line is both open and closed!
(
a
,
b
), then
a
(
a
c
)
c
(
c
b
)
9 Sketch graphs of continuous functions
f
:(
a
,
b
)
R for which the
rangeis
(i) a singleton point,
(ii) a closed interval,
(iii) an open interval,
(iv) a half-open interval,
(v) an open half-ray,
(vi) a closed half-ray,
(vii)
R
.