652 
+ Be, N)--+...= BN), then the set B(N.) + B(N.) +. 
also belongs to ‘this category, so that the complementary set cl = 
is everywhere dense. Now give «, NV arbitrarily. Let e >er and 
N<N; All the points of C(B) lie outside B (ez, N;), so that for 
every point of C(B) between N; and N'(er, N;) an index n; can 
be determined satisfying 
| F (2) — Sn, @)| Se Se 
Hereby the proof is ae since C'(B) is everywhere dense, 
so that the theorem of § 1 applies here. 
4. We also obtain a sufficient condition by substituting in the 
fore-going for B(e N): a nullset. For, a set which consists of a 
countable number of such sets is a null set, so that its complemen- 
tary set is everywhere dense and the fore-going reasoning applies 
again. It follows from this in particular that: a convergent series of 
continuous functions represents a continuous function if the conver- 
gence is “almost everywhere” quasi-unitorm. 
Il. 
5. In § 1 use has been made of the convergence of the series at 
the arbitrarily assumed point x, also in the case where w did not 
belong to the dense set H. The question may be put if it is necessary 
to suppose the series convergent throughout the whole interval. 
If a series of functions which are continuous throughout the 
interval a<a<bh converges uniformly at the points of a set # 
which is everywhere dense within this interval, then this involves 
the uniform convergence of the series throughout the whole interval. 
By analogy we are led to the following question: If the terms of 
a series are continuous functions of # in the intervala << 6, and 
if, besides, the series is quasi-uniformly convergent at the points of 
a set # which is everywhere dense within the said interval, is it 
then allowed to conclude to the convergence of the series throughout 
the whole interval and thereby to the continuity of the function 
represented by the series and thus to the quasi-uniformity of the 
convergence within a<aw<b? 
The answer is negative. In order to show this we consider the 
following series : 
f(@)=2—e4+ 2? —2?....-a7—ar+.... 
This series converges quasi-uniformly to zero within the open 
interval 0< 2< 1, which constitutes an every where dense set within 
the closed interval 0< «<1. The convergence is quasi-uniform, since 
