419 
: ; att DE 
vd, and y, Sd, the inequality Pyle vj) ZS 5 8 satisfied. 
Since f(x) tends to a finite limit as wv —0 there is a number d, 
such that for v7, << d, and y,< 4, the inequality Das Y,)) Sn is 
satisfied. 
If d< d, and at the same time < d, then 
‚Pp (ze, y) — En (es y,) | zal € for ©, <<. d and Y, xa d. 
There now exist only a finite number of indices for which Ed 
yv, 2d. At each of the points Br corresponding there to an index 
N, > N can be found such that 
en KE 
At the points of the considered set the sequence of functions 
DP, therefore converges quasi-uniformly. 
2. Conversely, let it be given that at the points of every set 
Are) Later Ya) a where OX ed, ry Sa, lim v= 0, 
k= 
hm y, = 0, this sequence of functions converges quasi-uniformly. 
k=a 
Let M denote the maximum of f(w) at 0, m the minimum. It is 
possible to construct a set of points z,,2,,... where x, >>0 and 
lim v= 0, such that Zon J (#,) = WM; similarly we can construct a 
(——— OO == 0) 
set Yi, Yay---, Where y, > O and my, = 0, such that lim fy) == nt 
: k=oa k=co 
If we now consider the set of points 
ENE BEN ese azen then (2 y,) = M—m, (1) 
k= 
unless m=-+ o or MM = — ow. Now it is possible, by hypothesis, 
to choose at each of the points P, from a finite number of indices 
1, 2,...n(s) an index n, such that 
€ 
|P (a3 yv) PT Py, (es y,) | < Di r F $ 5 5 (2) 
where « denotes a given arbitrary positive number. In consequence 
of the continuity of each of the functions ®, (v,y) and since the 
number of indices here considered is finite, a number Jd exists such 
that at every point P,, where v,<d and y, Cd the relation 
é 
eur, 1 ke en 
| Pr, (0, 0) — Ba, (erv) | < a 
is satisfied. 
Hence since 
€ 
D,, (00) = 9, wehave |, (vvv) | <5 - - + + @) 
