### Some properties of arbitrary functions

Fundamenta Mathematicae, Vol. 22 (1934), 70-76

Introduction

The object of this paper is to derive various properties of arbitrary functions of two and of three real variables concerning limits of functional values as a straight line or a plane is approached. As the point of departure the following theorem of Professor Blumberg is used:

*If f(x,y) is an arbitrary real function defined in a plane π; s a straight line in π; and d*_{1}, d_{2}, two directions of approach to s on the same side of it, then, for every point P of s with the possible exception of ℵ_{0} *points, I*_{Pd1} overlaps or abuts I_{Pd2}, where I_{Pd1} is the interval whose end points are the limits inferior and superior of f as (x,y) approaches P along the direction d_{1}, and I_{Pd2} is defined similarly.

Theorems 1 and 2 are generalizations of this theorem in different directions. First the effect upon the exceptional set of the neglect of various sets of points will be shown. Then the two given directions will be freed of their fixed positions to give a rather striking property of functions of two real variables. Finally an extension will be made to functions of three variables.

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