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
.
 
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