Mathematics. — “On the quasi-uniform convergence”. By Prot. 
J. Worrr. (Communicated by Prof. L. E. J. Brouwer). 
(Communicated at the meeting of February 22, 1919). 
I. 
We consider within an interval a<a<b6 a convergent series of 
continuous functions 7 (a) = f(a) + file) + .... 
As ArzELA has shewn a necessary and sufficient condition that 
J (x) should be continuous is the “quasi-wniform convergence’ of the 
series’). This serves to express that, if two positive numbers ¢ and V 
are assumed, ¢ as small as we please and N as large as we like, 
there exists a number N'>> N such that for each w of the interval 
a number n, of terms of the series can be determined between 
N and N', the sum of which, Su), differs less than e from f(«). 
This theorem constitutes, it is true, a complete solution of the 
problem: to replace the ordinary uniform convergence by another 
condition which is not only sufficient, but also necessary for the 
continuity of f(«). However the quasi-uniform convergence can again 
be replaced by a wider condition by which a slight extension of 
ArzeLa’s theorem is obtained. We shall namely prove the following 
theorem : 
1. If the series is quasi-uniformly convergent at the points of a 
set E which is everywhere dense within the interval a<ax<b, then 
f(@) is continuous throughout this interval. 
According to this supposition there exists for every e and every 
N a number N'> N such that for every « of FE an index n, can 
be determined (V << nz << N') such that | f(@)—Sn,(@)| Ze 
Now choose an arbitrary point « of the interval. In consequence 
of the convergence of the series a number MN can be found such 
that, s denoting an arbitrary positive number: 
AO en (D) | COU Oe UIN eo Se (i) 
For every n between N and the number V' conjugated to te, N, 
we can now in consequence of the continuity of S,(#) determine 
1) Mem. R. Acc. Bologna 1899. 
Boren, Legons sur les Fonctions de variables rééelles. 
