or Y1(X0 - X1) + Y2(X1 - X2) + ... + Y_n(X_(n-1) - X_n) + Y_n X_n.
Now, by supposition, all the numbers Y1, Y2 ... Y_n are finite, and
all the numbers X_(r-1) - X_r are of the same sign, hence by a known
algebraical theorem the series is equal to M(X0 - X_n) + Y_n X_n,
where M is a number intermediate between the greatest and the least of
the numbers Y1, Y2, ... Y_n. This remains true however many partial
intervals are taken, and therefore, when their number is increased
indefinitely, and their breadths are diminished indefinitely according
to any law, we have
_ _
/ b _ / b
| [f](x)[phi](x)dx = {[f](a) - [f](b)} M + [f](b) | [phi](x) dx
_/ a _/ a
when M is intermediate between the greatest and least values which
[int][a to x] [phi](x) dx can have, when x is in the given integral.
Now this integral is a continuous function of its upper limit x, and
therefore there is a value of x in the interval, for which it takes
any particular value between the greatest and least values that it
has. There is therefore a value [xi] between a and b, such that
_
_ /[xi]
M = | [phi](x)dx,
_/ a
hence
_ _ _
/ b /[xi] / b
| [f](x) [phi](x) dx = {[f](a) - [f](b)} | [phi](x) dx + [f](b) | [phi](x) dx
_/ a _/ a _/ a
_ _
/[xi] / b
= [f](a) | [phi](x) dx + [f](b) | [phi](x) dx.
_/ a _/[xi]
If the interval contains any finite numbers of points of discontinuity
of [f](x) or [phi](x), the method of proof still holds good, provided
these points are avoided in making the subdivisions; in particular if
either of the ends be a point of discontinuity of [f](x), we write
[f](a + 0) or [f](b - 0), for [f](a) or [f](b), it being assumed that
these limits exist.
_Functions, with Limited Variation._--The condition that [f](x), in
the mean-value theorem, either never increases or never diminishes as
x increases from a to b, places a restriction upon the applications of
the theorem. We can, however, show that a function [f](x) which is
finite and continuous between a and b, except for a finite number of
ordinary discontinuities, and which only changes from increasing to
diminishing or vice versa, a finite number of times, as x increases
from a to b, may be expressed as the difference of two functions
[f]1(x), [f]2(x), neither of which ever diminishes as x passes from a
to b, and that these functions are finite and continuous, except that
one or both of them are discontinuous at the points where the given
function is discontinuous. Let [alpha], [beta] be two consecutive
points at which [f](x) is discontinuous, consider any point x1, such
that [alpha] <= x1 <= [beta], and suppose that at the points M1, M2
... M_r between [alpha] and x1, [f](x) is a maximum, and at m1, m2 ...
m_r, it is a minimum; we will suppose, for example, that the ascending
order of values is [alpha], M1, m1, M2, m2 ... M_r, m_r, x1; it will
make no essential difference in the argument if m1 comes before M1, or
if M_r immediately precedes x1, M_(r-1) being then the last minimum.
Let [psi](x1) = [[f](M1) - [f]([alpha] + 0)] + [[f](M2) - [f](m1)]+ ...
+[[f](M_r) - [f](m_(r-1))] + [[f](x1) - [f](m_r)];
now let (x1) increase until it reaches the value (M_(r+1)) at which
[f](x) is again a maximum, then let
[psi](x1) = [[f](M1) - [f]([alpha] + 0)] + [[f](M2) - [f](m1)] + ...
+ [[f](M_r) - [f](m_(r-1)] + [[f](M_(r+1)) - [f](m_r)];
and suppose as x increases beyond the value M_(r+1), [psi](x1) remains
constant until the next minimum m_(r+1) is reached, when it again
becomes variable; we see that [psi](x1) is essentially positive and
never diminishes as x increases.