520 
Mr Young, On Monotone Sequences 
On Monotone Sequences of Continuous Functions. By W. H. 
Young, Sc.D., F.R.S. 
[j Received 2 February, 1908.] 
§ 1. In the following note I obtain the necessary and sufficient 
condition that a function should be representable as the limit of 
a monotone sequence of continuous functions, the word “ con- 
tinuous ” being used in the generalised sense, when + oo is 
distinguished from — oo . The result obtained is that for a 
diminishing sequence the function must be upper semi-continuous, 
while for an increasing sequence the function must be lower 
semi-continuous. 
In § 2 I shew that this condition is necessary. The method 
and notation here adopted are those of my paper on “Uniform 
and Non-uniform Convergence and Divergence of a Series of 
Continuous Functions and the Distinction of Right and Left,” 
Proc. L. M. S., Series 2, Vol. VI., pp. 29 — 51. I also use the 
conception of “ uniform divergence at a point ” given on p. 36, 
line 13, where, it should be noted, the sequence of functions con- 
sidered is a sequence of continuous functions. The definition of 
uniform divergence there given is only a suitable one in this case. 
In | 3 I give a generalisation of the result of § 2. 
In 4 and 6 I shew that the results of §§ 3 and 5 hold for any 
number of independent variables. 
In § 5 I shew that this condition is sufficient. 
In § 7 I give a new definition of a Riemann integral of a dis- 
continuous function, bounded or unbounded, based on the theorems 
of §§ 2 and 3. 
§ 2. Theorem. If f (x), f 2 (x), ... is a series of continuous 
functions ( bounded or not), having a definite limit f (x), finite or 
infinite , at each point x, and such that 
f(oo)^f 2 (x)^f(x)^ 
then (1) the function f is upper semi- continuous, and 
(2) there are no invisible points* of non-uniform convergence 
or divergence, in fact 
(3) Xl O) = ^ l 0) ^ t t l ( x ), 
=/(*)• 
A similar pair of relations holds, of course, on the right. 
* So that the function / is discontinuous wherever the series is non-uniformly 
convergent or divergent. 
