520 
DERIVATES, EXTREMES, VARIATION 
since x lies in (£ m . Thus BP' < 0 at any point of 21. Thus P is 
a monotone decreasing function in 21, by 508, 2. Hence 
P(c) — P (d) > 0. 
lienee /(c) - f{d~) - {Q (c) -Q(d)\> 0, 
or using 1), 3) 
p - 9 < 0, 
which is not so, as p is > q. 
2. (Lebesgue.) Let f(x), 9 (%) be continuous in the interval 21, 
and have a pair of corresponding derivates as Bf, Bg 1 which are 
finite at each point of 21, and also equal, the equality holding except 
possibly at a null set. Then f(x) — g(x) = constant in 21. 
The proof is entirely similar to that of 515, 3, the enumerable 
set (§ being here replaced by a null set. We then make use of 1. 
518. Letf(x) be continuous in some interval A = (u — S, u -j- &)■ 
Letf"(x) exist, finite or infinite, in A, but be finite at the point x — u. 
nen f («) = lim Qf. (1 
h—Q 
wJl6T6 
Qf(u) =/^ + K) , h> 0. 
Let us first suppose that/^/w) = 0. We have for 0< h<rj<b, 
\ i/Q + h') -f(u) f(u-h) -/00] 
Qf ~h{ h -hi 
= I ’ u<x'<u + h , u — h <x" <u 
h 
= y[(V — W )i/ ,, ( M ) + € '\ — ( x '' — u )lf"( u ) + «"i]» 
h 
where |e / 1, | e /; | are < e/2 for rj sufficiently small. 
Now x' — u^^ \x" —w_[<i 
h ~~ h ~ ’ 
while /"(w) = 0 , by hypothesis. 
Hence | (?/| < e , for 0 < h <rj, 
and 1) holds in this case.