1907-8.] Dr W. H. Young on a Test for Continuity. 
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XV. — On a Test for Continuity. By W. H. Young, Sc.D., F.R.S. 
Communicated by J. H. Maclagan Wedderburn. 
(MS. received February 20, 1908. Read March 16, 1908.) 
§ 1. The usual method of proving that a function defined as the limit 
of a sequence of continuous * functions is continuous is by proving that 
the convergence is uniform. This method may fail owing to the presence 
of points at which the convergence is non-uniform although the limiting 
function is continuous.]* In such a case it would be necessary to apply 
a further test, e.g. that of Arzela (“ uniform convergence by segments ”). 
In some cases the continuity may be proved directly by means of a 
totally different principle, without reference to modes of convergence at all. 
It is, in fact, a necessary and sufficient condition for the continuity of a 
function that it should be possible to express it at the same time as the 
limit of a monotone ascending and of a monotone descending sequence J of 
continuous functions. 
This test may or may not, of course, be an easy one to apply in any 
particular problem, but in certain cases, both in theory and practice, the 
application is almost immediate. 
The object of the present note is to call attention to the test, and show 
how it may be applied in a few simple cases. 
§ 2. For instance, take the continued fraction 
u Y u 2 
v \~ >r v ^ Jr 
where the u’s and the v’s are essentially positive continuous functions. 
* Throughout this paper the word “ continuous ” will be used to mean “ bounded and 
continuous” unless the contrary is stated. 
+ A point of non-uniform convergence where the limiting function is continuous is what 
I call an “ invisible point of non-uniform convergence.” 
+ /i j A j • • • • are said f° form a monotone ascending sequence if 
a . . . . , 
A , / 2 , . . . being functions of any number of variables. The theorem on which the test 
depends is that “ the limit of an ascending sequence of continuous functions is a lower semi- 
continuous function, and that the limit of a descending sequence of continuous functions is 
an upper semi-continuous function. A function which is the limit of both an ascending 
and a descending sequence is therefore both lower and upper semi-continuous, i.e. it is 
continuous.” It may be added that there are no invisible points of non-uniform convergence, 
or divergence, in the case of monotone sequences of continuous functions. — W. H. Young, “ On 
Monotone Sequences of Continuous Functions,” Carnb. Phil. Soc. Proc ., Lent Term, 1908. 
