Vol,. 8, 1922 
MATHEMATICS: H. BLUMBERG 
287 
By means of Lemma II, we prove also 
Theorem VII. The set of points P at which f is inexhaustibly approached 
and for which a non-vanishing partial neighborhood exists via which f is 
exhaustibly approached constitute a set of measure zero. 
Definition, /is said to be "neglectably approached at the point P = {^,r]) 
via M" if a sphere 5 exists with rj, /(^, rj)) as center, such that the pro- 
jection upon the XY plane of the set of points common to 5 and the surface 
z = f{x, y) has a set of measure zero in common with M. 
The following theorem generalizes Theorem VI. 
Theorem VIII . Let z = f {x, y) be any real, one-valued function defined 
in the XY plane. Then the points P of the XY plane that possess a non- 
vanishing partial neighborhood via which f is neglectably approached at P 
constitute a set of zero measure. 
Definition, f is "quasi-continuous"^ at P if for every e the set of 
points Q for which |/(0) — f{P) \ < e has 1 as exterior metric density at P. 
We have the following theorem, which generalizes theorem VIII. 
Theorem IX. f is quasi-continuous except at the points of a set of measure 
zero. 
Concluding Remarks. — As in the case of the descriptive properties, the 
metric theorems may be extended to many- valued functions. Theorem 
VIII, for example, when thus generalized reads as follows : Let z = f{x, y) 
be any real, single- or many -valued function defined in the entire XY 
plane. Then the points {x, y) of the XY plane for which a surface point 
/(^» y)) ci^d a non-vanishing partial neighborhood exist such that 
(x, y,f{x, y)) is neglectably approached via constitute a set of zero measure. 
The metric properties hold for functions of a single variable and, in 
general, for functions of n variables. Extension to space, to function 
space and to more general spaces would require a satisfactory definition of 
measure for such spaces;"^ it is not our purpose in this paper to enter 
upon such questions. 
Instead of projecting the surface points oi z = f{x, y) upon the XY 
plane, we may project them upon the X-axis and thus obtain other prop- 
erties. For example, let us define the relationship ^rmnn — between 
/-regions and points of the X-axis — as follows : / ^nrirzn ^ if the surface 
points having ^-coordinates in / have a limit point in the rectangle x = ^ 
n ^ y ^ r2, rz z S. n- ^nr2rsn is closed. By applying Lemma II 
to this closed relation, we obtain the following result: Let z = f{x, y) be 
any single- or many-valued function defined in the entire XY plane. Let 
^be a point of the X-axis of the following character: a surface point -q, ^) 
and a partial non-vanishing {linear) neighborhood of ^ exist such that 
^> r) ^'^ '^ot a limit point of surface points having x-codrdinates in . 
The totality of points ^ is of measure zero. 
Similar results may be obtained for other metric properties and also in 
