MAXIMA AND MINIMA 
parabolas P x , P 2 . The resulting graph defines a continuous func 
tion fix') which has proper extremes at the points Gs = 
However, unlike Ex. 1, the limit point x = 0 of these extremes is 
also a point at which f(x) has a proper extreme. 
Example 3. Let {¿if be a set of intervals which determine a 
Harnack set ¿p lying in 21 = (0, 1). Over each interval 8 = (a, /3) 
belonging to the n th stage, let us erect a curve, like a segment of 
a sine curve, of height h n = 0, as n = oo, and having horizontal 
tangents at a, /3, and at y, the middle point of the interval 8. At 
the points {f { of 21 not in any interval 8, let/(V) = 0. The func 
tion/ is now defined in 21 and is obviously continuous. At the 
points {yf,/ has a proper maximum; at points of the type a, /3, 
/has an improper minimum. These latter points form the set 
§ whose cardinal number is c. The function is increasing in each 
interval (a, y), and decreasing in each (y, /3). It oscillates in 
finitely often in the vicinity of any point of 
We note that while the points where / has a proper extreme 
form an enumerable set, the points of improper extreme may form 
a set whose cardinal number is c. 
Example u£. We use the same set of intervals {¿if but change 
the curve over 8, so that it has a constant segment r\ = (A, /x) in its 
middle portion. As before /=0, at the points | not in the 
intervals 8. 
The function/ (x) has now no proper extremes. At the points 
of £>, / has an improper minimum ; at the points of the type X, /a, it 
has an improper maximum. 
Example 5. Weierstrass’ Function. Let (5 denote the points in 
an interval 21 of the type 
x — 
r, s, positive integers. 
For such an x we have, using the notation of 502, 
b m x = i m + = b m ~ s r. 
f m = 0 , for m>s. 
<W=(-l> +1 = (-iy +1 . 
Hence 
Thus