PROCEEDINGS 
OF THE 
NATIONAL ACADEMY OF SCIENCES 
Volume 8 OCTOBER 15, 1922 Number 10 
NEW PROPERTIES OF ALL REAL FUNCTIONS 
By Henry Blumberg 
D^JPARTMENT OF MATHEMATICS, UNIVERSITY OF I1.I.INOIS 
Communicated June 10, 1922 
In a former paper ^ the author communicated a number of properties of 
every real function /(x), which were stated in terms of the successive 
saltus functions associated with a given function. The present results 
do not involve these saltus functions and are direct qualifications 
of f{x). Since f{x) is entirely unrestricted, except, of course, that 
it is defined — even this restriction may be partially dispensed with — ■ 
and therefore finite for every real x, these qualifications are consequences 
of nothing else than that f{x) is a function. A new light is thus thrown 
upon the nature of a function. 
The new properties are of two kinds, descriptive and metric ; the former 
are concerned with density and the latter with measure (Lebesgue). 
For the sake of greater concreteness, we discuss, for the most part, 
planar sets and real, single-valued functions of two real variables. 
1. Descriptive Properties. — We say that a planar set 5 is an "/-region" 
( = open set) if every point of 5 is an inner point of 5. We deal with 
binary relations 9^ between /-regions and points. i9?P shall mean that 
the /-region / has the relation to the point P. The relation is said 
to be "closed" if the relationships I^Pn and lim Pn = P imply I^P. 
n — >- 00 
By a "neighborhood" of a point P, we understand an /-region containing 
P; by a "partial neighborhood" of P, an /-region of which P is an inner 
or a boundary point. 
We have the following 
Lemma I. If ^ is a closed relation, then the points for which (a) NdlP for 
every neighborhood N of P, and {b) a partial neighborhood N^ exists such that 
A/'< 9^ P (i.e., N^ dlP is false) constitute a non-dense (i.e., nowhere dense) set. 
An important example of a closed relation appears in connection with 
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