250 
Proceedings of the Royal Society of Edinburgh. [Sess. 
The odd convergents form a monotone descending sequence having the 
continued fraction as limit, so that the latter is an upper semi-continuous 
function. The even convergents form a monotone ascending sequence 
having the continued fraction as limit, so that it is lower semi-continuous, 
and therefore, being also upper semi-continuous, is a continuous function, 
and this whatever the number of variables on which the u’s and the v’s 
depend. 
§ 3. Again, the u’s having the same meaning as before, the infinite 
series 
zq - zq -I- zq - zq + . . . . , 
if the sum at each point is definite, necessarily represents a continuous 
function when the u’s form a monotone descending sequence. In fact, it 
is the limit of the ascending sequence of functions constituted by the 
partial sums of the series 
(zq - zq) + (zq - zq).+ .... . . . (L) 
and also of the descending sequence constituted by the partial sums of the 
series 
zq - (zq - zq) - (zq - zq) - . . (2) 
It should be noted that in the case of such a series “ term-by-term 
integration ” is allowable. In fact, each of the series (1) and (2) is so 
integrable, since it is monotone and represents a continuous function. This 
is therefore the case with the original series. 
§ 4. If we omit the condition that the series 
zq ~ U 2 + • • • * 
should have a definite sum at each point, the monotone ascending sequence 
of continuous functions f l , f 2 , . . . . , where 
fn=(u 1 -U 2 ) + (u B -U 4 )+ .... + - M 2n ) , 
defines a limiting function f which is > u x — u 2 and is lower semi-continuous. 
On the other hand, the monotone descending sequence of continuous 
functions g x ,g 2 .... , where 
9n ~ — (^2 — ^ 3 ) — • • • • (^2w— 2 ^2w— l) j 
defines a limiting function g which is < u x and is upper semi -continuous. 
Moreover, since 
fn y Qn i 
zq - zq < / < g < zq (1) 
Thus at each point P there are two possible values for the sum of the 
series, according as we proceed to the limit by taking an even or an odd 
