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Proceedings of the Poyal Society of Edinburgh. [Sess. 
The argument of § 3 shows, in fact, that if the series (3) converges, the 
series (1) represents a continuous bounded function in the closed interval 
(0 , r). 
If the series (3) diverge to the value + oo , the series (1) still represents 
a continuous function f(x) from 0 to r, the latter not included, by the 
preceding article. At each point x < r we have 
f(x) = (a 0 - ap) + (a 2 x 2 - a 3 x 3 ) + . . (4) 
while the right-hand side of this equation, being the limit of a monotone 
increasing sequence of continuous functions, represents a lower semi- 
continuous function in the closed interval (0, r). The value of this lower 
semi-continuous function at r being, however, + oo , the function is con- 
tinuous there, so that the series represents a continuous but unbounded 
function f(x) throughout the closed interval (0 , r). 
Similarly, if the series (3) diverge to the value — oo , using the grouping 
(2) of the preceding article, and the fact that an upper semi-continuous 
function is continuous where it has the value — oo , we prove the continuity 
of f{x) in the closed interval (0, r). 
Finally, let the series (3) oscillate between the limits of indeterminancy 
L and U. Then, as in the case when the series (3) had the value 4- oo , we 
get the equation (4), whose right-hand side represents a lower semi- 
continuous function in the closed-interval (0 , r). The value of this function 
at r, being one of the values the series (3) is capable of assuming, is > L. 
But this value is <c the limits of the values of the lower semi-continuous 
function in the neighbourhood, that is, is < the limits of f(x) as x approaches 
r. This shows that all such limits are >L. Similarly, using the grouping 
employed when the series (3) had the value — oo , we show that all 
such limits are <U. Thus all such limits lie between L and U, both 
included. 
§ 7. We now proceed to give two examples, the first constituting another 
example of the application of § 3 ; and the second, of the original test of § 1 . 
Example 1. — Consider the integral 
uo/)= 
J 0X + l 
dx 
where y is any positive quantity. This may be written 
Jo+f;+ • • • • = u i(y)~ u M+ • • • • 
Here the u’s are essentially positive and continuous functions of y and 
form a monotone descending sequence. Thus, by § 3, U(^/) is a continuous 
