1907-8.] Dr W. H. Young on a Test for Continuity. 257 
for all values o/ x< a and each value of y in a certain neighbourhood of y, 
J“u(x , y)dx is continuous at the point y = y 0 . 
As an example of the application of the last test, we see that, fix , y) 
being a positive function continuous with respect to the ensemble (x , y ), if 
J" /(as , y)dx and J"/(cc , y) sin x . dx are both convergent at y = y Q , and the 
former is a continuous function of y at y 0 , so is the latter. 
Addendum (received 9th March 1908). 
§11. The theorem quoted in § 5 (footnote), on which the test given in 
that article depends, is a particular case of the following more general 
theorem : — * 
“A monotone decreasing (increasing) sequence of f unctions f Y , f 2 , . . . . 
which are upper (lower) semi-continuous at P has for limit a function/ 
which is also upper (lower) semi-continuous at P.” 
This latter theorem also may, of course, be used as a test for continuity 
at a definite point. The necessary and sufficient test for continuity at a 
point P will then be the possibility of expressing / both as the limit of a 
monotone decreasing sequence of functions which are upper semi-continuous 
(or in particular continuous) at P, and of a monotone increasing sequence of 
functions which are lower semi-continuous (or in particular continuous) at P. 
§ 12. It should be noticed that the methods given above may be still 
further generalised. We have seen that to prove a function continuous 
throughout an interval, it is sufficient to show that it is the limit both of a 
monotone increasing and of a monotone decreasing sequence of continuous 
functions. The theorem quoted in the preceding article shows thatYo prove 
that a function is pointwise discontinuous, it is sufficient to show that it is 
the limit both of a monotone increasing and of a monotone decreasing 
sequence of pointwise discontinuous functions. 
The theorem quoted, however, may be taken to apply not only to a 
fundamental interval, but to any fundamental perfect set ; hence, if it is 
desired to show that a function is pointwise discontinuous with respect not 
only to an interval, but with respect to every perfect set in that interval, 
it is sufficient to show that it is the limit both of a monotone increasing 
and of a monotone decreasing sequence of functions with the same property. 
It is a simplification in this case, however, that, in applying the test, 
we may use semi-continuous functions. To prove that a function is upper 
(lower) semi-continuous, we only need to show that it is the limit of a single 
VOL. XXVIII. 
* Loc. cit. ; for proof see Mess, of Math., 1908. 
17 
