284 
MATHEMATICS: H. BLUMBERG 
Proc. N. a. S. 
a function. Let z = f{x, y) be any given one- valued function of the two 
real variables x and y, and let / be defined for the entire XY plane. Let 
9^ = 9^rir2> where n < f2 are two real numbers, be defined as follows : Idlnn P 
if and only if an infinite sequence of points P„ of the /-region I exists such 
that lim P„ = P, lim / exists, and n^Hm f{Pn)^r2', here P, P„ = 
n 00 n — >- 00 n ->- oo 
(:^„, yn) and/(P),/(P„) = /(^, y„). 
Definition. The function /(jt:, y) is said to be "densely approached at 
the point r?)" or in other words, the point 77, /(^, 77)) of the "surface" 
z = f{x, y) is said to be densely approached, if for every positive e there 
exists a planar neighborhood N of ry) such that the points of N for which 
|/(^> 3^) ~ fi^j v) \ < ^ form a dense set in N. 
Definition. An "exhaustible" set (= set of first category, according 
to Baire) is the sum of i^o non-dense sets; a "residual" set, the comple- 
ment (with respect to the XY plane) of an exhaustible set.^ 
By the aid of Lemma I and the closed relation ^rmy we prove 
Theorem I. For every real function f{x, y) whatsoever, the points of the 
surface z = f{x, y) that are densely approached form a residual set. Con- 
versely, given any residual set R whatsoever, a function f{x, y) exists that is 
densely approached at and only at the points of R. 
The following definition of dense approach is equivalent: The function 
f(x, y) is said to be densely approached at P = (^, r]) if, for every partial 
neighborhood of P, the set of points of the surface z = f{x, y) correspond- 
ing to the points of has r), /(^, r?)) as a limit point. By the use of this 
definition, we get the following theorem, which is equivalent to Theorem 
I, but which shows better, perhaps, the remarkable degree of "microscopic 
symmetry" an unconditioned function possesses. 
Theorem 1'. With every function f{x, y) whatsoever, there is associated a 
residual set R — dependent on f — of the XY plane such that if P = 77) is 
a point of R and N ^ , a partial neighborhood of P, then (^, 77, /(^, 77)) is a limit 
point of the set of points {x, y, f{x, y)) for which {x, y) is in . 
Definition. The function / is said to be "inexhaustibly approached" 
at the point P if every neighborhood of P contains, for every e > 0, an 
inexhaustible set of points — i.e., a set that is not exhaustible — at which / 
differs from /(P) by less than e. 
If M is any planar set, we use, in connection with approach, the ex- 
pression "via M" to designate that {x, y) is restricted to range in M. 
Thus "/ is inexhaustibly approached at P via M" means that for every 
neighborhood N oi P and every e > 0, the set MN, which is the aggregate of 
points common to M and N, contains an inexhaustible set of points at 
which / differs from /(P) by less than e. 
We have the following 
Theorem II. For every function f{x, y), there exists in the XY plane a 
