Since
and
, the functions
and
are
-chains if
and
are.
The second assertion is obvious since
.
Define
by
where
and
are positive real numbers. The boundary of
is easily seen to
be
.
Given
, let
where
is the function of
Problem 3-41 extended so that it is 0 on the positive
-axis.
Let
so that
is an integer because
.
Define
.
One has
and
. On the other
boundaries,
and
.
So
, as desired.