518 
DERIVATES, EXTREMES, VARIATION 
For let 
Then in 
<t>=f-g » ÿ=g -/• 
•A = 3i - (S, 
120' > Ä/' — Rg' = 0 , Äty' > 0. 
But if Rcf>' < 0 at one point in 3i, it is < 0 at a set of points 33 
whose cardinal number is c. But 33 lies in (g. Hence H(f) is 
never < 0, in 31. The same holds for yjr. Hence, by 508, <£ and 
yjr are both monotone increasing. This is impossible unless 
cf> = a constant. 
516. The preceding theorem states that the continuous function 
f(x) in the interval 31 is known in 31, aside from a constant, when 
/' (x) is finite and known in 31, aside from an enumerable set. 
Thus f(x~) is known in 31 when /' is finite and known at each 
irrational point of 3i. 
This is not the case when/' is finite and known at each rational 
point only in 31. 
For the rational points in 3f being enumerable, let them be 
(1 
Let 
r v r 2’ r 3 •” 
I = l\ + 4 + 
be a positive term series whose sum l is < 31. Let us place r x 
within an interval S r of length < l x . Let r L be the first number 
in 1) not in 8 r Let us place it within a non-overlapping interval 
8 2 of length < Z 2 , etc. 
We now define a function f(x) in 31 such that the value of/at 
any x is the length of all the intervals and part of an interval 
lying to the left of x. Obviously f(x) is a continuous function of 
x in 31. At each rational point /' (x) = 1. But f(x) is not de 
termined aside from a constant. For < l. Therefore when 
l is small enough we may vary the position and lengths of the 
5-intervals, so that the resulting/’s do not differ from each other 
only by a constant. 
517. 1. Let fix) be continuous in 31 = (a < b) and. have a finite 
derivate, say Rf, at each point of 31. Let (5 denote the pomts of 21 
where It has one sign, say >0. If (S exists, it cannot be a null set.