286 
MATHEMATICS: H. BLUMBERG 
PrOc. N. a. S. 
defined for 0 ^ ^ 1. The ecart between fi{x) and f^ix) is defined to be 
max|/i(:r)-/2W|. 
The assumption that / is single-valued may also be dropped without 
invalidating Theorems I and II — Theorem III, of course impUes single- 
valuedness by its very nature. We thus get the following generalization 
of Theorem I (and a similar one for Theorem II). 
Theorem I Let f{x, y) be any real function defined for the entire XY 
plane and taking at every point at least one value — the number of values may 
change, however, from point to point and vary from i to c, the cardinal num- 
ber of the continuum. Then the points {x, y) such that every surface point 
{x, y, f{x, y)) is densely approached by the surface z = fix, y) constitute a 
residual set. 
3. Metric Properties. — As in the case of Section 1, we discuss in this 
section planar sets and functions of two variables. 
Let S be any planar set; P, a point of 5; C^, a circle with P as center and 
r as radius; m{Cr), the area of and me{SCr), the exterior lycbesgue meas- 
ure of the portion of 5 in C^. Then if 
r o m{Cr) 
exists and is equal to k, we say that the "exterior metric density" of 5 at 
the point P is ^. We have the following 
Theorem V.^ Let S be any planar set. Then the points of S at which the 
exterior metric density of S is 9^ 1 — ^i.e., the points where the exterior 
metric density either does not exist or does exist and is less than 1 — 
constitute a set of zero measure {Lebesgue). 
Definition. N ^ is said to be a ''non-vdnishing partial neighborhood of 
P," if the exterior metric density of N ^ at P is ^ 0. We have the following 
lemma, which corresponds to Lemma I for the descriptive properties : 
Lemma II. Let ^ be a closed relation as in Lemma I. The points P for 
which (a) NdlP for every neighborhood of P and (b) a non-vanishing partial 
neighborhood exists such that diP (i-e., P is false) constitute a 
set of zero measure. 
By the aid of this lemma we prove 
Theorem VI. Let f{x, y) be any real, one-valued function defined in the 
entire plane. Then there exists in the XY plane a set Z — dependent on f — 
of measure zero, such that if (a) (x, y) is any point of the XY plane not be- 
longing to Z; (6) N^, any non-vanishing partial neighborhood of {x, y); and 
(c) S, any sphere with {x, y,f{x, y)) as center; then there is at least one point 
of the surface z — fix, y) lying in the sphere S and having as projection upon 
the XY plane a point in . 
This theorem becomes false if we omit the restriction that the partial 
neighborhood N ^ shall be non- vanishing. 
