424 
2, <a. For V we now choose an arbitrary set of points P,(z,,2,, 2), 
Perio Es SUG Zonen Where DF Zor UF Zo ie Ep 
Wms} 
Amy, = 2e 
== 
Ex hypothesi it is possible at every point P, of V to choose x 
such that 
k 
€ 
Pog ptp I — WEISS 2 @ 
Since yw is continuous and the number of indices n, is finite a 
belted ! ih Am di | De 
positive number d exists such that for eld and |y,—z,|<d 
we have 
| € 
Un, (x, yy 2) eee Wn. (2. Zo Ze) | = at AE (5) 
SINCE Yn, (2, 2,2) = 0, it follows from (4) and (5) that 
| w (©. Uy 20) |< & for | TZ Sand | hinted 
I (#,) == ze) fw) (ge) | 
ies Ts | Se 
amd Om oa 
q ; fe) — FG) eet 7 
It results from this, that tends to a finite limit as 
Ay == 
k 0 
wv F2 coincides successively with the points of an arbitrary 
set of which z, is the sole limiting point, and from this again 
that a finite lim OL ee) exists, hence f(z) has a finite derivative 
Tame ete Ze 
at z,. Since this holds good for any ,;z| <a it follows from a 
theorem enunciated by Goursar that f(z) is analytical within |z| <a. 
It may be stated that here only a part of the supposition has 
been made use of. 
8. If the sequence of functions f, (2), f, (2)... each of which 
is analytical for |z) <a, converges at a point z, internal to this 
7 J) za 7 AW) 
converges quasi- 
Y 
uniformly at the points of every closed set internal to |x| <a, lyl <a, 
then f (2) converges for every |z| <a, the limiting function f is 
region and the function f* (w, y) 
wv 
analytical and everywhere there is 
J'42) Shin ph (z). 
n= 00 
That i (z) converges everywhere is involved in the convergence 
