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.

~ 146. 147.

Differential quotient of the integral.,

255

or likewise:
b a+-(b —a) b
Jf(x) p(x) dx = f(a) f(p(x)d x + f(b) jp (x) d x-)
a a a -O(b -a)
Note. The values of f at the extremities a and b can also be
indeterminate. From our deduction it follows, that then in case of
a decreasing function, f(a) is to be replaced by the greatest value and
f(b) by the least value to which this function approximates in the
neighbourhood of these points; the reverse holds in the case of an
increasing function.
147. Having learned that the definite integral is a continuous
function of its upper limit; we naturally ask, whether its progressive
and regressive differential quotients have a single determinate value?
From the Equation VIII.:
x +h x x +h
IF(x+ )-F(x)= (x) d f(d(x) dx =f(x) lx = h(g + (G —))
a a x
we find:
F(x + h) - F(x)
+ =g + e (G -g).
Now making h converge to zero, it is evident.that wherever the
progressive value of f(x) is continuous, where we can therefore put
g + O(G- g)-'f(x + Oh),
the function F(x) has the progressive differential quotient
F'(x) f(x + 0);
and wherever the regressive value of f(x) is continuous, the function
F(x) has the regressive differential quotient f(x - 0).
The same holds also at every point at which f(x) differs from
the values f(x + 0) and f(x - 0) by an arbitrary finite quantity;
and there may be a discrete set of such points, provided, arbitrarily
near each such point, the values f(x + 0) and f(x - 0) follow from
f(x + h) and f(x - h) by continuous transition. As a particular case
we have under these hypotheses:
b b
d SJf(x) dx == Lim - ff(x)dx = f(b).
a x=-b b-h
Regarding the integral as a function of its lower limit, the differential
quotient can be determined either by means of the inversion of limits:
b a
J f(x) dx - ff(x) dx,
a b
*) Du Bois-Reymond, Journal f. Mathem., Vol. 69.