MATHEMATICS: H. BLUMBERG 
647 
It is seen that 
s{g,x) = u{g,x) - l{g,x). 
Let us, for the sake of brevity, write 
s(g, x) ^ s'(x), s(s\ x) ^ s'^ix), sis\ x) ^ s'^'ix), .... 
Sierpinski^ has proved the 
Theorem. s"{x) = s"\x) = ^''(x) .... 
The chief aim of the present paper is to communicate a number of 
companion propositions to Sierpinski's theorem. The new results are 
based on the definition of other types of saltus, which immediately 
suggest themselves and arise from the one described above, when cer- 
tain specified subsets of the range of the independent variable may be 
neglected. 
The first new type arises when finite subsets may be neglected. For 
every subinterval {a, (3) of (a, b), there evidently exists a number, which 
we denote by Uf{g, a^), uniquely characterized by the following double 
property:^ For every e >0, on the one hand, the set of points of the 
interval (a, ^) for which g(x) > Uf{g, a^) + e is finite, while on the other 
hand, there exists an infinite set of points of (a, /3) at which g{x) > 
Uf{g, a^) — €. This number Uf{gy a^) is the lower-bound of all possible 
upper-bounds that g{x) may have in the interval (a, jS), in case a finite 
set of points may be neglected. Likewise, there is a number, which we 
denote by lf{g, a^) characterized by the property that for every e > 0, 
the set of points of (a, /3) where g{x) < lf{g, a(3) — e is finite, whereas, 
the set of points of {a, (3) where g{x) < lf{g, a/S) -f e is infinite. lf{g, a^) 
is the upper-bound of all possible lower-bounds of g{x) in (a, jS), when a 
finite number of points may be neglected. Finally, we denote by 
Sf{g, cc^) the lower-bound of the saltus of g{x) in (a, jS), in case a finite 
number of points may be neglected. Evidently 
Sf{g, a(3) = Uf{g, a^) - lf{g, a/3). 
We shall designate the numbers 
Uf(g, a(3), l/{g, a^) and Sf{g, a0) 
as Hhe f -upper-hound,^ 'the f -lower-bound' and Hhe f-saltus' of g(x) in the 
interval (a, (3). As in the case where no point may be neglected, we now 
define 'the f -upper-bound' 'the f -lower-bound/ and 'the f-saltus' of g(x) 
at the fixed point x of {a, b), as the lower-bound of Uf{g, a^), the upper- 
bound of lf{g, a/S) and the lower-bound of s^ig, a^) for all possible sub- 
intervals (a, /3) of (a, b) that contain x as mtenor pomt. With the 
given function g{x), we have thus associated 'the f -upper-bound func- 
tion,^ 'the f-lower-bound function' and 'the f-saltus function' which we 
denote by 
