Definition: We say that the discontinuities of a function of $(x, y)$ are regularly distributed in $S$ or in $T$ if they all lie on a finite number of curves with continuously turning tangents, no one of which is met by a line parallel to the axis of $x$ or of $y$ in more than a finite number of points. 1. Some Preliminary Propositions and Definitions.
In order to avoid interruptions in later sections, we collect here certain propositions of the integral calculus for future reference. We shall have to deal with functions of one and of two variables. The independent variables, which we will for the present denote by x and (x, y) respectively, are in all cases real. In fact, in order to avoid unnecessary complications we will assume that, unless the contrary is explicitly stated, all quantities we have to deal with are real. The range of values of the single argument x is usually
I a <= x <= b.
We shall speak of this in future simply as the interval. In the case of functions of two variables, two cases have to be considered. Interpreting (x, y) as rectangular coordinates in a plane, we sometimes consider the square:
S a <= x <= b & a <= y<= b
and sometimes the triangle
T a<=y<=x<=b
It should be noticed that the three regions we have just defined, I, S, T, are closed regions, that is they include the points of their boundaries.
In order to avoid long circumlocutions we lay down the following: We say that the discontinuities of a function of (x, y) are regularly distributed in S or in T if they all lie on a finite number of curves with continuously turning tangents, no one of which is met by a line parallel to the axis of x or of y in more than a finite number of points.
In order to make the enunciation of some of our results simpler, we will assume once for all that the functions we deal with are defined even at the points of discontinuity, at least in the cases where they remain finite in the neighbourhood of such points. The following theorem will be important for us. We state it first for the case of the region S.
THEOREM 1.
If the two functions f(x, y) and g(x, y) are finite in S and their discontinuities, if they have any, are regularly distributed the function b F(x,y) = INT (f(x,v)g(v,y)dv a is continuous throughout S.
The truth of this theorem becomes evident if we interpret (x,y,v) as rectangular coordinates in space. It is then clear that the function under the integral sign is finite throughout the cube
a <= x <= b, a <= y<= b, a <=v <= b,
and becomes discontinuous in this cube only at points on two sets of cylinders whose generators are parallel respectively to the axes of x and y. Moreover these cylinders are so shaped that any line x = xo, y = yo in this cube meets them at only a finite number of points. The formal proof, based on these or similar considerations, presents no difficulty, and we leave it for the reader.