An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

Sixth Chapter.
General theorems concerning the definite integral as the limiting
value of — a sum.
142. The fundamental problem of the Integral Calculus in its
simplest statement (~ 101) leads to the evaluation of the limiting
value of a sum with arbitrarily many summands. Independently
therefore of the differential conception, the problem of the integral
calculus opens up the question: What must be the nature of a function f(x) in the interval from x = a to x = b, in order that the sum:
S= df(a + O d) + d2f(x, + 02d2)+ d3f (x2+Q,3d3)+-.. +dnf(x,-l+Ondn)
may have a determinate finite limiting values when the subdivision of
the interval from a to b by the points:
x1 = a + d1, x2 = xx2 - X3 x 2+d3..., = xn-_ + dn
is continued arbitrarily, while the lengths d converge to zero? Such is
the most general form in which this question can be proposed. The
quantities 0 denote proper fractions, they may also be zero or unity,
so that the values of the function are always chosen anywhere within
or at the limits of an interval. The limiting value must be quite
independent of the arbitrary quantities 0. Still more generally we
may denote by f(x - OQ dp) any value whatever from the greatest to
the least of the values assumed by the function in the interval dp.
If it be discontinuous in the interval, this selected value may not
occur among those of the function. It is a secondary question that
must be answered by itself, whether, when there is a limiting value,
this limiting value regarded as a function of the upper limit has f(x)
as its derived function or not.
Now while in ~ 102 the investigation admitted of a simple form,
because f(x) was assumed continuous, it will have to be conducted
differently here, since we have first to ascertain the hypotheses
necessary regarding the function t(x). Riemann*) who was the first
*) Riemann: Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe (Werke, pp. 213-253). Some details in the following proof have
been rendered more precise by Du Bois-Reymond (J. f. M., Vol. 79).