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function (a more general notion of function than had been considered
in the eighteenth century) as the limit of
f
(
x
)(
x
x
) as thelengths
of thesubintrvals
tend to 0. Although Cauchy presumed
uniform continuity for his functions, the definition enabled him to prove
the Fundamental Theorem of Calculus as we know it. In the same
lectures, Cauchy obtained the Taylor series with integral form of the
remainder. In 1822 Fourier's work on the conduction of heat raised the
possibility of the integrability of discontinuous functions obtained from
trigonometric series. Cauchy acknowledged that his definition of the
integral could be applied to a function with a finite number of
discontinuities and gave formal definitions for improper integrals.
Dirichlet who had discussions with Fourier, and perhaps Cauchy, in
Paris published his famous example (qn 11) of a function which is
not
equal to a Fourier series and is
not
integrable in Cauchy's sense in 1829
and asserted in the same paper that a function was integrable in
Cauchy's sense provided its discontinuities were nowhere dense.
Riemann tackled the subject of what functions could be integrable
under the direction of Dirichlet, and in 1854 prepared a lecture in
Go¨ ttingen (which was not published until 1868, after his death) in
which the possibility of the integration of trigonometric series was
considered to a depth that was to be definitive for the rest of the
century. Riemann's definition of integral was the limit of
f
(
t
x
x
x
t
x
)(
x
), where
x
, as the lengths of the subintervals
x
tend to 0, which is equivalent to Cauchy's definition but
without the restriction to continuous functions, and he constructed a
function which was integrable in this sense but which was
discontinuous on a set of points that was everywhere dense. We have
given an example of such a function in qn 36 (using a construction due
to Thomae, 1875). Riemann's example was put together rather
differently: let
g
(
x
)
0 when
x
[
x
]
x
and let
g
(
x
)
x
[
x
]
otherwise, then Riemann's function
g
(2
x
)
2
g
(3
x
)
3
...
g
(
nx
)
n
...
f
(
x
)
g
(
x
)
which is discontinuous at every point
x
p
/2
q
, where
p
and
q
areodd
numbers without a common divisor. The function
f
is nonethelss
integrable because the series is uniformly convergent everywhere.
The treatment of the Riemann integral that we have given, with
step functions, upper and lower sums, and upper and lower integrals is
due to Darboux (1875). Darboux showed that the integral of Riemann's
function was continuous and not differentiable at the points where the
original function was discontinuous.
