Vol. 8, 1922 
MA THEM A TICS: H. BL UMBERG 
285 
residual set R — dependent on f — such that if P is a point of R, and , a 
partial neighborhood of P, the function fis inexhaustibly, and therefore densely 
approached at P via RN^ . 
With the aid of Theorem II we prove 
Theorem III. With every function f{x, y) there is_ associated — not 
uniquely, however — a dense set D of the XY plane, such that fis continuous if 
{x, y) ranges over D. 
2. Generalizations. — The considerations and theorems of Section 1 
apply not only to functions of n real variables, but to every space 5 that 
satisfies the following four conditions : 
(1) 5 is metric;^ that is to say, with every pair of elements P and Q 
of S there is associated a non-negative, real number PQ (Frechet's ecart) 
in such a way that if P, Q and R are three elements of 5, then 
(a) PQ = QP ; 
(b) PQ = 0, when and only when P = Q; and 
{c)PQ + QR^PQ. 
(2) 5 is complete (vollstandig) ;^ that is to say, II \ ri, r2} ■ • ■ } I^n • • • J 
is a "regular" sequence of elements of 5— in other words, for every e >0 
there exists an integer n^ such that PxP,, < e for '^>n^ and > n^ — there 
exists a limit element P (i.e., an element P with the property lim P„P = 0). 
(3) 5 contains a denumerable subset that is dense in 5. 
(4) 5 has no isolated points. 
We may thus state the following theorem — the definitions of the terms for 
S may be obtained after slight and evident modifications from those 
for the plane. 
Theorem IV. Let S be any ^complete, metric space containing a dense, 
denumerable subset and without isolated points; and f{P), any real function 
defined for the elements P of S. Then there exists a residual set R, such that 
if P is a point of R, and N ^ a partial neighborhood of P, the function f is 
inexhaustibly and therefore densely approached at P via RN^ . Also there 
exists a dense subset D of S such that f (P) is continuous if P ranges over D. 
As particular examples of a complete, metric space, with a dense, de- 
numerable subset and without isolated points, we mention : 
(a) Euclidean n-space where the ecart between two points is the euclidean 
distance between them. 
(6) A perfect subset of euclidean space. 
(c) Hilbert space, that is, the ensemble of sequences (xi, Xo, ... x„, . . .) 
n = 00 
of real numbers with convergent l^Xn"^. The ecart between two "points" 
w = l 
(xi, X2, ... x„, . . . ) and {yi, y^, ... y^, . . . ) is defined to be 
^{xr-yiY^{x2-yiY^ ... . 
{d) Function space: .S consists of all real, continuous functions f{x) 
