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(i) How do you know the sequences ( a
) and ( b
) areboth
convergent?
(ii) If ( a
)
A and ( b
)
B , why must f ( A )
0
f ( B )?
(iii) Why must A B and f ( A ) 0?
Wesummarisethersult of qn 13 by saying that, if
f :[ a , b ] R is a continuous function, f ( a ) 0, f ( b ) 0, there
exists a c ( a , b ) such that f ( c ) 0.
14 By applying the result of qn 13 to a suitably chosen function, prove
that, if f :[ a , b ] R is a continuous function and f ( a ) k f ( b ),
then there exists a c
( a , b ) such that f ( c )
k .
15 By applying the result of qn 14 to a suitably chosen function, prove
that if f :[ a , b ]
R
k f ( b ),
is a continuous function and f ( a )
then there exists a c ( a , b ) such that f ( c ) k .
Questions 14 and 15, together, give the general form of the
Intermediate Value Theorem, that a continuous function f :[ a , b ]
R
takes every value between f ( a ) and f ( b ).
R
16 Givean exampleof a discontinuous function f :[ a , b ]
which
takes every value between f ( a ) and f ( b ). The existence of such a
function shows that the converse of the Intermediate Value
Theorem (qn 13) is false.
17 Provethat a continuous function f : R Z is necessarily constant.
18 Provethat a continuous function f : R Q is necessarily constant.
19
(i) For any real numbers a and b show that the function defined
on
x ax b has a real root. To what class of
polynomial functions may this result be extended?
(ii) For any positive real number a and integer n
R
by f ( x )
1, show that
thefunction f defined by f ( x ) x a has a positiveroot. Use
this to provide an alternative proof of the existence of n th
roots for positive numbers to that given in qn 4.40.
20 If f : [0, 1] [0, 1] is a continuous function, by applying thersult
of qn 13 to an appropriately chosen function, prove that there is at
least one c [0, 1] such that f ( c ) c . Givean exampleto show that
this result might fail if the domain of f were (0, 1). Generalise this
result for a continuous function
f :[ a , b ]
[ a , b ].
21 Can the Intermediate Value Theorem be used to prove that the
image of an interval under a continuous function is necessarily an
interval? Or, in other words, under a continuous function, is a
connected set always mapped onto a connected set?
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