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10 Give examples of monotonic functions
f
and
g
which areboth
continuous on [
a
,
b
], for which theequation
(
f
g
)
f
g
would fail if the integral were always to denote the positive value of
the area between the graph and the
x
-axis.
A further distinction between the notion of integral and the
measurement of area under a graph seems natural when we recognise
that the argument of qn 7, as modified before qn 10, does not depend
on thecontinuity of thefunction
f
. An area must have a boundary, and
in order to give
boundaries
to the areas under discussion in qn 7, the
function
f
was taken to be continuous. The graph of a discontinuous
function may have gaps in it and so does not bound an area. But the
limiting arguments of qn 7 as modified before qn 10 hold for a
monotonic function
f
even if
f
is discontinuous at many points of the
interval [
a
,
b
]. So such a function can havea well-defined integral.
11 (
Dirichlet
, 1829) If even discontinuous functions can have integrals,
are there any functions which cannot? Find the area of the smallest
circumscribed rectangle and the largest inscribed rectangle for the
area bounded by
x
0, the
x
-axis,
x
1 and the function defined
by
1
0
when
x
is rational,
when
x
is irrational;
f
(
x
)
on theintrval [0, 1].
As we search for a definition of
f
, wewill limit our attention to
functions which are bounded, so that they may be covered above and
below, by rectangles, and we seek a definition with the following two
properties which we retain from our experience of finding areas with
curved boundaries using inscribed and circumscribed polygons:
(i) if
m
f
(
x
)
M
on [
a
,
b
], then
m
(
b
a
)
f
M
(
b
a
);
(ii)
f
f
f
, when these integrals exist.
Step functions
In qns 12
—
15, weconstruct a family of functions, called
step
functions
, with convenient, and obvious, integrals, which we define in qn
