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a
A
, and if further we suppose that
f
is differentiable at
a
, and
g
is
differentiable at
f
(
a
), usethechain ruleto provethat
1
g
(
f
(
a
))
(
a
)
.
f
The Leibnizian expression is again particularly suggestive here:
dx
dy
1
dy
dx
.
41 If
f
and
g
areinvrsefunctions as in qn 40, and
f
(
a
)
0, provethat
g
cannot be differentiable at
f
(
a
).
Indicate by a sketch why this should be expected and relate your
answer to qn 5.
42 Let
f
bea continuous bijction
f
:
A
B
where
A
and
B
areopen
intervals, and let
g
:
B
A
betheinvrseof
f
. Wesaw in qn 7.25
that
g
is also a continuous bijection. Now suppose that at some
point
a
A
,
f
is differentiable and
f
(
a
)
0.
(i) Provethat
g
(
f
(
x
))
g
(
f
(
a
))
f
(
x
)
f
(
a
)
1
f
(
a
)
.
lim
(ii) Now let
f
(
a
)
b
, and let (
b
) be a sequence in
B
which tends
to
b
, but which does not contain any term equal to
b
.
Why must there exist a unique
a
A
,
a
a
, such that
?
(iii) Usethecontinuity of
g
to provethat (
a
f
(
a
)
b
)
a
.
(iv) Usethelimit aboveto provethat
g
(
b
)
g
(
b
)
1
f
(
a
)
,as
n
.
b
b
(v) Deduce that
g
is differentiable at
f
(
a
).
The only definition or theorem in this chapter which depends on
the Completeness Principle is qn 42. Completeness is needed for part
(iii). This was necessary because, on rational domains, differentiable
bijections do not have to have continuous inverses. See Ko¨ rner (1991).
43 If thefunction
f
is defined by
f
(
x
)
0 and a positive
integer
n
, identify the inverse function
f
, and find its derived
function.
If, for positive
x
, and positive integers
p
and
q
, wedefine
x
for
x
x
(
x
)
,
