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attain its bounds when examined on the domain [0, 6]. Does this
contradict the result of the previous question?
The result of qns 31 and 32 is usually described by saying that a
continuous function on a closed interval is
bounded
, and thersult of qn
34 by saying that a continuous function on a closed interval
attains its
bounds
. Questions 31 and 34(i)
—
(viii) givethe
Maximum theorem
.
Questions 32 and 34(ix) give the
Minimum theorem
. Although both
results depend on completeness it is not necessary for the Intermediate
Value Theorem to precede their proof. Taken with the Intermediate
Value Theorem, they imply, for continuous real functions, that a closed
interval is mapped to a closed interval or a point.
36 Give examples to show that continuous functions on unbounded
intervals or open intervals or half-open invervals need not be
bounded and, even if bounded, need not attain their bounds.
Uniform continuity
Both the sequence definition of continuity and the neighbourhood
definition of continuity define the continuity of a function at a
point
,
and weonly claim continuity on an
interval
when the function is
continuous at every point of that interval. Moreover, when determining
continuity at a point with the neighbourhood definition of continuity,
given an
may be different for continuity at a point
a
from what is needed to establish continuity at a point
b
. Weexamine
now, when, with a given
, a choiceof
may bemadewhich will
establish continuity throughout an interval. The roˆ le of closed intervals
again turns out to becritical.
, the necessary
37 If
f
:[
10,10]
R is defined by
f
(
x
)
x
, provethat if
x
y
1/20 then
f
(
x
)
f
(
y
)
1. Extend this result to show
that, for any positive
,if
x
y
/20 then
f
(
x
)
f
(
y
)
.
38 If
f
: (0, 1)
R is defined by
f
(
x
)
1/
x
,
f
(
x
) is unbounded.
Find a
such that if
x
, then
f
(
x
)
f
(
)
1.
Find a
such that if
x
, then
f
(
x
)
f
(
)
1.
Show that, if
x
a
implies
f
(
x
)
f
(
a
)
1, then
a
/(1
a
).
Deduce that there is no constant
such that
x
a
implies
f
(
x
)
f
(
a
)
1 for all
x
and
a
in thedomain of thefunction
f
.
A function
f
:A
is said to be
uniformlycontinuous on A
when,
given
0, there exists a
such that
l
R
l
l
l
.
x
y
f
(
x
)
f
(
y
)