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.

256 The definite integral as the limiting value of a sum. Bk, III. ch. VI.

leading to:
b
d jf(x)dx =-f(a);
a
or directly from the formula:
a + h
Lim - J- f (x)d = - f(a).
a
At all points in whose neighbourhood f(x) has infinitely many
maxima and minima with finite fluctuations, moreover such points also
can only form a discrete set, the function '(x) has no differential
quotient.
Summing up it may therefore be said: Every definite integral
expresses within its interval of integration a continuous function of
its rupper limit, whose progressive differential quotient has generally a
determinate value, and its regressive also one identical with this. It is
only in discrete sets of points that the progressive and regressive
differential quotients can differ or can be altogether indeterminate.
Since the definite integral is quite independent of the nature of
the function in discrete points, all integrable functions that coincide
within their interval except in such points give rise to the same value
of the definite integral.
From this appears, further, what is the form of the connexion
between the definite and the indefinite integral. For, if F(x) be any
continuous function, whose derived function F'(x) is integrable, (so
that if this derivate be discontinuous or indeterminate or else have
infinitely many maxima and minima with finite fluctuations it is so
only at infinitely many discrete points,) then, the difference:
x
F'(x) dxx - F(x)
a
is a continuous function of x, whose differential quotient, generally
zero, is primnd facie indeterminate only in discrete points. But such
a function is a constant. For in every interval however small,
after separating out the singular points, finite intervals will still
remain within which not only is the function continuous but also
its derived function vanishes. By the Theorem ~ 100 therefore the
function is constant within such an interval; and because the limits
of the interval can be brought arbitrarily close to the singular points,
since the function is continuous it will have the same value also at
the singular points. Therefore it does not undergo any change of