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.

268 The definite integral as the limiting value of a sum. Bk. III. ch. VI.
is a continuous function of a. We have:
IO W
j[/(a;, ~a 7) -' (x, a)]x = 'x h) - f(x, a)]dx
a a
co
+ Jt'fx, a ~ h) - f(x, a)]dx.
In order that by choice of h this expression may become smaller than Y,
the function must be so constituted that, for all values in the interval
from a - h to a + h, one and the same w shall be sufficient to make:
0o
j[f'(x, a + ) - f(x, a)] dx < d.
w
Thus ex. gr. (~ 155) for every finite value of a:
0
/ sin ax
but for a = 0 the value of the integral vanishes. The definite integral
is therefore not a continuous function of a, although the function
that is to be integrated is continuous in both variables.
But the hypothesis is fulfilled, if, whatever value a may have,
the function f vanish determinately for x = c in an order higher
than the first. For then we can first assume w so great as to make:
o00
[f(x, a + h) - f(x, a)] dx < ~,
w
and afterwards determine the value of h so that we may have:
j [/(xa ) -f'(x, a)] dx < ~
a
In this case differentiation under the integral sign is certainly
possible, if /'(x) also is a continuous function of both variables
that vanishes in higher than the first order for x - oo. For, putting:
f/x, a + h) - f(x, a) af(x, a+- o h)
h= a a 1
we have:
f'(xa + h) - f(x,a) __ )f(x, ] d f(x+, a Oh) __ f(Xa) dx)
J L h a j (x, a +: a ]f
a aa, r fxae 8f~xia)] clx.