251 
1907-8.] Dr W. H. Young on a Test for Continuity. 
number of terms in the partial summations ; both these values are finite, 
and the first is not greater than the second. 
If by any law we select one of these two values, or any intermediate 
value, at every point we get a function, say li, which may be said to be 
defined by the series, and we shall have at every point 
/ <. h < g (2) 
It now follows that, if at P the series have a definite sum, P is a point 
of continuity for each of the functions h defined by the series. For by (2) 
the least limit of h(x) as x approaches P as limit ;> the least limit of 
f(x), > /( P), since / is lower semi-continuous, > h( P). Comparing, on the other 
hand, with g, the greatest limit of h(x) < h{ P), so that h{x) has the limit 
A(P), i.e. h is continuous at P. 
§ 5. The argument used in the preceding article is of a perfectly general 
nature : it consists in the use of a special case of the following theorem, of 
which the proof * is now obvious : — 
Theorem . — The necessary and sufficient condition that a function h(x) 
should be continuous at P is that it should be 'possible to construct two 
monotone sequences of functions continuous at P, one ascending and one 
descending, having at P the same limit h(P), and such that, in the neighbour- 
hood of P, the limits of the ascending sequence are not greater, and those 
of the descending sequence not less, than the corresponding values of h(x). 
This theorem constitutes a test for continuity at a point. 
§ 6. A particular case of § 4 is when we have a series 
a 0 - apc4- a 2 x 2 - . . . . (1) 
where all the <x’s are positive and 
a 0 > ape, > a 2 x 2 > . . . . . . . . (2) 
for all positive values of x less than r, it being assumed that the series 
converges for all such values of x. 
We can then assert that the series represents a continuous function 
f(x) from 0 to r, the latter not included only when the series 
a 0 - a x r -f a 2 r 2 - . . . . (3) 
oscillates. In the latter case, moreover, any limit which f(x) can 
approach as x approaches r lies between the limits of indeterminancy of 
(3), both included. 
* It involves the use of the theorem that “ the limit of a descending (ascending) sequence 
of functions continuous at P is a function which is upper (lower) semi-continuous at P.” 
This, like the previous theorem quoted on p. 249, of which it is a generalisation, will he found 
proved in the paper in the Camb. Phil. Soc. Proc. above referred to. 
