CHAPTER XV 
DERIVATES, EXTREMES, VARIATION 
Derivates 
498. Suppose we have given a one-valued continuous function 
f(x) spread over an interval 21 — (a <b~). We can state various 
properties which it enjoys. For example, it is limited, it takes 
on its extreme values, it is integrable. On the other hand, we 
do not know I o how it oscillates in 2Í, or 2° if it has a differ 
ential coefficient at each point of 21. In this chapter we wish to 
study the behavior of continuous functions with reference to these 
last two properties. In Chapters VIII and XI of volume I this 
subject was touched upon; we wish here to develop it farther. 
499. In I, 363, 364, we have defined the terms difference quo 
tient, differential coefficient, derivative, right- and left-hand dif 
ferential coefficients and derivatives, unilateral differential coeffi 
cients and derivatives. The corresponding symbols are 
” , /'(«) , /'00 , Rf(a) , 
Ax 
Lf 00 , Rf O) , Lf O). 
The unilateral differential coefficient and derivative may be de 
noted by 
Uf'(a) , Uf (*). (1 
When 
lim A/? 
a=o Ax 
/'(«) = lim 
A=0 Ax 
does not exist, finite or infinite, we may introduce its upper and 
lower limits. Thus 
A/ , /(a)=limf£ (2 
A=0 ^ x 
always exist, finite or infinite. We call them the upper and lower 
differential coefficients at the point x = a. The aggregate of values 
493