648 
MATHEMATICS: H. BLUMBERG 
Uf{g, x), lf(g, x) and Sf(g, x) 
respectively. 
In the second place, the subsets that may be neglected are denumer- 
able. As before, there exists a number, which we denote by Ud{g, al3), 
uniquely characterized by the property that, for every e > 0, the set 
of points of (a, /3) where g(x) > Ud(g, afi) + e is denumerable, whereas 
the set of points where g{x) > Ud{g, a(3) — e is non-denumerable; this 
number Ud(g, a^) we call Hhe d-upper-bound^ of g(x) in {a, jS). Precisely 
as before we define the related numbers 
ld{g, a^), Sd{g, a/3), Ud(g, x), ld(g, x) 
and Hhe d-saltus function* Sd{g, x). 
In the third place, 'exhaustible^ sets (i.e., sets of first category^) may 
be neglected. We then obtain Hhe e-saltus function' Se{g, x), together 
with the related numbers. 
Finally, sets of (Lebesgue) zero measure may be neglected. We then 
obtain Hhe z-saltus function^ Sz{g, x), together with the related functions. 
As in the case where no point may be neglected, we write 
Sfig, x) = sj{x), Sf{sf, x) = Sf{x), Sf{s'/^, x) = sy\x), . . . ; 
and similarly for the J-saltus, the e-saltus and the z-saltus functions. 
Having defined the new types of saltus we had in view, we may now 
state the corresponding analogues of Sierpinski's theorem. The most 
interesting and least obvious results are the following two theorems. 
Theorem. s^ (x) = sjix) = sl(x) = . . . . 
Theorem. s^' {x) = sY(x) = sl(x) = . . . . 
Moreover, as examples show, s^ (x) and s^ (x) may be different from 
s'J'{x) and s'J\x) respectively. 
In the case of the f -saltus, the functions s/''\x) (n = 1,2, . . . ) may 
all be diferent. 
In the case of the e-saltus, while s^ {x) may be =±= s^ {x), we have 
Theorem. \x) = 0. 
A generalization of our result for the c^-saltus is as follows : 
Theorem. If s'^{x), s"^{x), s'^{x) . . . represent the successive saltus 
functions that arise when, instead of neglecting denumerable sets (i.e., sets 
of cardinal number So), 'we neglect sets of cardinal number i<, where 
is any cardinal number < c, the cardinal number of the continuum, then 
(*) - s^{x) ^ s]^{x) ^ . . . . 
Because of our negative result in the case of the /-saltus, we are nat- 
urally led to define sf\x) for transfinite /3's. Our result will show that 
it is sufiicient to confine ourselves to transfinite numbers belonging to 
Cantor's second class. If ^ is not a limiting number, /3 — 1 exists, and 
