546 
DERIVATES, EXTREMES, VARIATION 
point of 53. If W are the points in 53 which correspond to the 
points V in 51, then the tangent is parallel to the ¿-axis at the 
points IF, or Gr r (¿) = 0, at these points. The points IF are pan 
tactic in 53- 
Let Z denote the points of 53 at which Cr' (¿) = 0. We show 
that Z is of the 2° category, and therefore 
Card Z—c. 
For Gr' (f) being of class <1 in 53, its points of discontinuity S 
form a set of the 1° category, by 486, 2. On the other hand, the 
points of continuity of Gr' form precisely the set Z, since the 
points IF are pantactic in 53 and Gr' = 0 in IF In passing let us 
note that the points Z in SB correspond 1-1 to a set of points ,3 at 
which the series 3) diverges. For at these points the tangent to 
F is vertical. But at any point of convergence of 3), we saw in 
2 that the tangent is not vertical. 
Finally we observe that 3) shows that 
Min D(x) • — 2 ^a n , in 51. 
3 sip 
Hence 
Summing up, we have this result: 
8. Let the positive term series 2Va n converge. Let = be 
an enumerable pantactic set in the interval 51. The Pompeiu curves 
defined by 
F(x^)= 2a n (x-c n y 
have a tangent at each point in 51, whose slope is given by 
when this series is convergent, i.e. for all x in 51 except a null set. 
At a point set 3 of the 2° category which embraces (5, the tangents 
are vertical. The ordinates of the curve F increase with x. 
540. 1. Faber Curves.* Let F(x) be continuous in the interval 
5l=(0, 1). Its graph we denote by F. For simplicity let 
* Math. Annalen, v. 66 (1908), p. 81.