of Continuous Functions. 
527 
§ 6. The same is true, if, instead of one variable x, we under- 
stand by x the ensemble of two or more variables, i.e. if the point 
x, instead of being a point on a straight line, is a point in the 
plane, or in n-dimensional space. 
Suppose, for definiteness, that x is a point in a plane, and that 
the upper semi-continuous function f(x), which in case(l) is never 
negative, is defined for a finite closed region R. Enclose the 
region in an equilateral triangle, and assign to f at all points of 
this triangle outside R the value zero, which does not disturb its 
upper semi-continuity. 
This equilateral triangle we divide successively into 4, 4 2 , ... 
equal equilateral triangular parts, just as, on the straight line, 
we used equal segments. The primary points are now the vertices 
of the triangles, the extremes, those vertices which lie on the 
periphery of the large equilateral triangle, or fundamental triangle. 
At each primary point used at the nth division, and at each 
extreme point used at the nth division, we define f n , as before, 
as the upper bound of /in all the triangles meeting at that primary 
or extreme point. Erecting ordinates of height f n at the vertices 
of any triangular part, we lay through their three tops a plane, 
which serves, as before the straight line, to define f n at every point 
of that triangular part. The proof that the series of con- 
tinuous functions f n so defined is monotone and has f as limit is 
of precisely the same nature as before. 
In case (2) the value to be assigned to f outside the closed 
region R, is, not zero, but the upper bound of / in order that f 
may remain lower semi-continuous. 
Having in this way obtained the representation of a bounded 
semi-continuous function which is not negative for every point x 
of a triangle containing the closed region R, the same repre- 
sentation does for the region R itself, x being now a point of that 
closed region alone. This representation can then be applied in 
case (3), as on the straight line. The reasoning in case (4) is the 
same in all dimensions. 
The above indicates how the proof must be verbally modified 
when # is a point in a plane. In the case of S 3 we use, instead 
of a triangle, a regular tetrahedron, and in S n a regular (n + l)-gon. 
§ 7. The results of the previous articles enable us to give a 
definition of an absolutely existent integral (Riemann integral) 
which applies equally whether the integrand is bounded or un- 
bounded, whether the range of integration be a segment, or an 
area, or indeed any closed ^-dimensional region, whatever be the 
nature of its boundary. This definition is as follows : 
Form the associated upper limiting function cf> of the given 
function /. Express it as the limit of a monotone descending 
