MATHEMATICS: H. BLUMBERG 
649 
we define 5) (x) as equal to Sf{Sf ,x). For our purpose, therefore, 
all we have to do now is to define sf\x), in case |8 is a limiting number, 
in terms of the functions sf\x)y where v < ^. This we do simply by 
means of the equation 
where [vn] is a sequence of numbers less than /S. It is seen that this 
limit always exists and is independent of the particularly chosen se- 
quence {vn}. 
Our positive result for the case of the /-saltus may now be stated as 
follows : 
Theorem. There exists a number /3 of the first or the second class, 
such that 
Furthermore, it is shown by means of an example, that if jS is a given 
number of the first or the second class, then the functions sp(x) (1 
^ /8) may all be different, whereas sf\x) = sf'^^\x). 
The following interesting connection exists between the J-saltus and 
the /-saltus. 
Theorem. For every function g{x) for which sf\x) = Sj^'^^\x) 
and according to the preceding theorem there always exists such a ^ 
of the first or the second class , we have 
Sd(sf, x) = sf\x). 
The results may be easily extended to the case of many-valued, 
bounded, or unbounded functions of several variables, or of infinitely 
many variables; and by means of simple postulate systems, to more 
abstract situations. 
The above is essentially the introduction of a paper, which is to be 
offered to the Annals of Mathematics. 
^ Throughout the paper, we use the expressions, ' the upper-bound' and * the lower- 
bound,' in the sense of 'the least upper-bound' and 'the greatest lower-bound/ respec- 
tively. 
2 Of course, it will be understood, that in case x = a, b — and only then — we permit 
ot, jS to coincide with x. This remark applies also to similar situations below. 
3 The functions u{g, x) and l{g, x) are often, though not quite unobjectionably, called 
the 'maximum' and the 'minimum' functions belonging to g(x). Cf., for example, Hob- 
son, The Theory of Functions of a Real Variable (1907), art. 180. 
^Bull. Acad. Sci., Cracovie (1910), 633-634. 
5 Cf. Baire, Acta Mathematica, 30, 21 and 22 (1906). 
« Cf. Denjoy, /. math. Paris, Ser. 7, 1, 122-125 (1915). 
