255 
1907-8.] Dr W. H. Young on a Test for Continuity. 
Example . — 
Let f(x) = (1 - x)~ h in the closed interval (0 , 1) , 
and =(a?-l) -2 when x>l but <2. 
This function, being continuous but not bounded, and always positive, 
can be generated as the limit of such a sequence of functions as that used 
by De la Vallee-Poussin. We have, however, 
F(a?) = 2 - 2(1 -- x )° in the closed interval (0 , 1) , 
and = + oo , when x >1 but < 2 . 
It should be noticed, further, that the argument used in this article 
depends in no way on the form of the functions devised by De la Vallee- 
Poussin, but only on the fact that they form a monotone sequence. 
§ 9. We now deduce two subsidiary tests which are sometimes useful. 
The first applies to series and the second to integrals. Both follow 
immediately from the principle enunciated in § 1. They are analogous to 
certain known tests for uniform convergence, but more general in form. 
Theorem . — If 
u i + u 2 + • • • • 
is a series of continuous functions of any number of independent 
variables which is convergent at all points of a certain region, then it 
represents a continuous function provided we can find a second series 
U 1 + U 2 + .... 
where the U’s are positive continuous functions, whose sum is continuous r 
and are such that 
! uj | <Ui 
for all values of i and each point of the region considered. 
For the function in question is the limit of the monotone ascending 
sequence f lf f 2 , • • • • , where 
fn~ u l + u 2 + • • • • + U n - 1 “ - U n+1 - . . . . 
and is evidently continuous. 
It is also the limit of the monotone descending sequence g 1} g 9 , . . . ._ 
where 
9n = u 1 + ^ 2 + • • • • + u n - 1 + + U n+ i + . . . . 
and is evidently continuous. 
Hence the result follows. 
Theorem. — 7/u(x,y) is a continuous function of the ensemble (x, y), 
and J*u(x, y)dx is convergent for every value of y considered, and if we' 
