of Continuous Functions. 523 
being made of those which deal with the distinction of right and 
left. 
It follows that a monotone decreasing ( increasing ) sequence of 
continuous functions of m variables has as limiting function an 
upper {lower) semi-continuous function, whether the various functions 
concerned are bounded or not, and that there are no invisible points 
of non-uniform convergence or divergence. 
§ 5. Theorem. Conversely, if f{x) is an upper {lower) semi- 
continuous function, bounded or unbounded, it is representable 
as the limit of a decreasing {increasing) monotone sequence of 
functions which are continuous, but not necessarily finite, each of 
which may be so chosen as to have a finite loiver {upper) bound. 
Case 1. First let f {x) be upper semi-continuous, bounded, 
and never negative*. 
Divide the fundamental segment {a, b) in successive stages 
into 2, 4, 8, . . . equal divisions. The points of division used at one 
stage remain points of division at all subsequent stages, and 
ultimately become dense everywhere; they are called the primary 
points, and are -countable. The remaining points, other than the 
extreme points a and b, are called secondary points. 
If a; be a primary point used in the n\\i division, it is the 
end-point of two abutting parts, forming together a closed segment 
to which x is internal. We take the value of f n at x to be the 
upper bound of yin this segment. At a and b we take as value 
of f n the upper bound of / in the single part having the point in 
question as end-point. 
We have thus a finite positive value of f n {x) at a finite 
number of points. Erecting ordinates at these points of height 
= f n there, we join the tops of adjacent ordinates by straight lines. 
The continuous line so obtained is taken to define 
V =fn 0) 
at every point of the segment {a, b). 
Since the upper bound of / in a closed segment is not less than 
the upper bound of /in a closed segment contained in the first, we 
see that at each primary point, and at the extremes a and b, 
fn — i ^/f 
The same is then true at the secondary points, since the 
straight line joining two points on two ordinates lies above that 
* The case where / (;c) is bounded has been proved by Baire, cp. Hobson’s 
Functions of a Real Variable, where the proof given is substantially that used in 
case (1) above. The fact that the sequence obtained is monotone does not seem 
to have been mentioned, and had no particular interest in connection with Baire’s 
researches. The form of the proof given above avoids the use of an e, and is 
required for subsequent generalisation. 
