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an interval (w—d,r + d) such that for every point § within it the 
inequality 
SON sO 
holds good. Since the number of these indices» is finite, an interval 
TI exists, with centre z such that for any & within / the relation (2) 
is satisfied by any nm between N and N’. Thus, if § is a point of 
E lying within /,and if for » is chosen the index n: corresponding 
to £, in the first place the relations (1) and (2) are satisfied and 
besides 
Sn. (RACLETTE PS a. (8) 
It follows from (1), (2), and (8) that 
| F(e) —F(O) |< 
Hence we have f(@=limf ©, where § coincides successively 
Er 
with all the points of the everywhere dense set /. If «' is a point 
of / not belonging to #, then 
f(@)—F(e) | =lin| F@ —7O| Se 
Hereby the continuity of f(x) is established. 
2. In connection with Arzrra’s theorem it appears thus that the 
quasi-uniform convergence at the points of a set £ everywhere 
dense within a Sr <5 involves quasi-uniform convergence throughout 
the whole interval. 
3. From the fore-going it may be easily concluded that the 
quasi-uniform convergence in ARZELA'’s theorem can be replaced by 
the following criterion: “for every ¢, NV there exists an N’ > N 
and a set of points B(e, NV), belonging to the first category of 
Barre, such that for every « of the interval not belonging to B (e, N) 
an index nm, (N<n,< N’) can be determined which satisfies 
FO Sne’. 
In order to establish this we take provisorily a fixed number M 
and a decreasing series of positive numbers ¢,,¢,,&,.... having 
zero for limit. Let N'(e,%, N) and B(er, N) correspond to en with 
the above-mentioned meaning. Since B (ej, N) consists of a countable 
set of nowhere dense sets of points, this is also the case with 
the set B(e,, N) + Bie,, N) -+....= B(N), 80 that B (N) also belongs 
to the first category of Barrw. Now choose an increasing sequence 
tending to infinity and put B(e,, Vj) + 
43%" 
of numbers N,, N,.... 
