522 Mr Young , On Monotone Sequences 
we get the remaining relation in (3), viz. 
/(P) = 7T i (P). 
From this it follows also that 
/(P)^<MP), 
which shews that f is an upper semi-continuous function on the 
left. 
The relation (3) shews that if the upper and lower associated 
functions </>£ and yjr L coincide with f, the peak and chasm functions 
7 tl and % L are equal, in other words, at a point of continuity on 
the left there is uniform convergence on the left, or which is the 
same thing, there are no invisible points of non-uniform con- 
vergence on the left. 
Similar relations holding of course on the right, this proves 
the whole theorem. 
§ 3. Bearing in mind that the investigations of my L. M. S. 
paper, referred to above, apply at a point at which the generating 
functions fi,f>, ••• are all continuous, and further that any count- 
ably infinite set of pointwise discontinuous functions defined for 
any interval have in common an everywhere dense set of common 
continuities, the proposition in the text may be generalised as 
follows : 
Theorem*. Let f 1} f 2 , ••• be a decreasing monotone sequence of 
upper semi-continuous functions having f for limiting function, 
then f has in every interval a point of upper semi-continuity. 
In fact the second part of the preceding argument applies, as 
it stands, at a common point of continuity of f, f,, ..., if we bear 
in mind that the upper semi-continuous function f n (f) necessarily 
assumes its upper bound in (P, Q), by the well-known property of 
upper semi-continuous functions. 
| 4. So far we have assumed that the functions concerned are 
functions of a single variable. We have, in fact, treated the left 
and right separately. There is, however, nothing to prevent our 
dropping the suffixes L and R, and defining the various functions 
introduced in the argument in such a way that the proof applies 
to functions of m variables. The intervals terminating at the 
point P will be replaced by regions surrounding the point P. 
The various theorems of my L. M. S. paper, above referred to, 
hold, in fact, for functions of any number of variables, exception 
* This theorem gives only part of the truth. The function / is indeed upper 
semi-continuous throughout the interval or region. Moreover if/ 1 ,/ 2 , ••• are upper 
semi-continuous at a point P only and from a decreasing sequence in its neighbour- 
hood, then / is upper semi-continuous at P. See my paper “On Functions 
Defined by Monotone Sequences, and their Upper and Lower Bounds,” Mess. Math. 
New Series, No. 442 (1908), p. 151. 
