MAXIMA AND MINIMA 
521 
Suppose now that f" (u) — a =£ 0. Let 
g(x) =f(x) — q(x) , where q(x) = % ax 2 + bx + c. 
Since q" (u) = a , g"(u) — 0. 
Thus we are in the preceding case, and lim Qg — 0. 
But Qg=Qf-Qq- 
Hence lim Qf = a. 
Maxima and Minima 
519. 1. In I, 466 and 476, we have defined the terms f(x) as 
a maximum or a minimum at a point. Let us extend these terms 
as follows. Let f (x 1 ••• x m ) be defined over 91, and let x= a be an 
inner point of $1. 
We sag f has a maximum at x = a if l°,/(a) —/(#)> 0, for any 
x in some V(a), and 2°, f (a) —f (pc) > 0 for some x in any V(a). 
If the sign > can be replaced by > in 1°, we will say f has a 
groper maximum at a, when we wish to emphasize this fact; and 
when > cannot be replaced by >, we will say f has an improper 
maximum. A similar extension of the old definition holds for 
the minimum. A common term for maximum and minimum is 
extreme. 
2. If f(x) is a constant in some segment 53, lying in the inter 
val 91, 53 is called a segment of invariability, or a constant segment 
of/in 91. 
Example. Let fix') be continuous in 9l = (0, 1*). 
Let 
x = • a.aM, 
l tt 2 № 3 *’* 
(1 
be the expression of a point of 91 in the normal form in the dyadic 
system. Let 
% — ' **1*2^3 *"* 
(2 
be expressed in the triadic system, where « n =a n , when a n = 0, 
and =2 when a n — 1. The points = form a Cantor set, 
I, 272. Let |3ns the adjoint set of intervals. We associate