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.

~ 145. 146. Examples of integrable discontinuous functions.

249

a finite interval there are only a finite number of fractions having
denominators below a given finite limit.
The series converges uniformly; it is not continuous, because its
separate terms are not continuous functions; the integral is obtained
by integration of the separate terms.
146. The fundamental theorems concerning the definite integral
follow immediately from the equation of definition, whose shortest
form, independent moreover of the quantities 0, is:
b
j f(x) dx = Lim { (x - a)f(a) + (x, -- xI) (x)* * * + (b-x,,_-) f (xn-_) }.
a
We have:
b b
~I. f ~cf(x)dx cJ f(x)dx.
Ca a
II. Interchanging a with b, and keeping the same partition of the
interval, the integral:
a
J f(x) dx
b
is obviously equal to:
Lim { (x,_l-b)f(b)+(Xn-2-Xn-l)f(Xn-l)+-'-(xl-X2)/Z2)+(a-xl)f(Xl) }
and also to:
Lim { (xn-l - b)f(xn-1) + (x_.2 -Xn-xl)f(Xn-2) + * - (x - x2)f(xI)
+ (a- x)f'(a)}.
For, as was shown, it is indifferent at what points within an interval
the values of the function are chosen.
It becomes evident by this second equality that:
b a
Jf(x) dx = f (x)dx,
i. e. the integral changes only in sign by interchanging its upper and
lower limits.
c b b
III. Jif(x) (x + Ji(x) dx 1= J'(x) dx.
aG C a
This equation holds even when c lies outside the interval a to b,
provided only the function remains integrable. For, when:
a < b <c
we have: