646 MATHEMATICS: H. BLUMBERG 
TABLE 1. 
Lowering of the Freezing-Point in a Mixture of Mannite and Potassium Chloride 
CONC. MANNITE 
CONC. KCl 
MIXTUilE 
FREEZING-POINT LOWERING 
DEVIATION 
Mannite 
KCl alone 
0.00493 
0.01071 
0.02183 
0.04067 
0.00987 
0.02153 
0.04367 
0.08134 
0.04479 
0.09676 
0.19400 
0.35700 
0.00927 
0.02012 
0.04061 
0.0757 
0.03557 
0.07670 
0.15343 
0.2813 
0.00005 
0.00006 
0.00004 
0.0000 
So far as the writers have been able to find, this is the first work on 
mixtures of salts which has been carried out with an accurate temper- 
ature measuring system. One set of measurements on mixtures of a 
salt with a non-electrolyte has been carried out by Osaka [Zs. physik. 
Chem.j 41, 560 (1902)], but his results show a greater deviation than 
that illustrated by the preceding table. 
We wish to express our indebtedness to the National Academy of 
Sciences for a grant from the Wolcott Gibbs Fund, which was used for 
the purchase of the temperature measuring system, to W. P. White for 
the design of a special potentiometer, and to L. H. Adams and John 
Johnston of the Geophysical Laboratory of the Carnegie Institution for 
the loan of the freezing-point apparatus. 
CERTAIN GENERAL PROPERTIES OF FUNCTIONS 
By Henry Blumberg 
DEPARTMENT OF MATHEMATICS. UNIVERSITY OF NEBRASKA 
Received by the Academy. October 14, 1916 
Let g{x) be a real, one-valued, bounded, continuous or discontinuous 
function defined in the interval (a, b). The functional values of g{x) 
in the subinterval {a, /3) of (a, b) have a least upper-bound, a greatest 
lower-bound and a saltus, which we denote respectively by 
u{g, a^), l{g, al3) and s{g, a^) = u{g, a^)- l(g, a/3). 
The upper-bound, the lower-bound^ and the saltus of g{x) at the fixed 
point X of (a^ b) are defined respectively as the (greatest) lower-bound 
of u(g, a/3), the (least) upper-bound of l(g, a:/3) and the (greatest) lower- 
bound of s(g, q;/3) for all possible subintervals (a, /S) of (a, b) that contain 
X as interior point. ^ With the given function g(x), there are thus asso- 
ciated three new functions of x, the upper-bound function, the lower- 
bound function and the saltus function, which we denote by 
u(g, Kg, and s{g, x) 
respectively.^ 
