421 
lf a sequence of functions f («) is given each of which is 
« Le l 7 t 
continuous for OSa«<a and has a finite derivative Ff, ©) at 0, 
Onde ih for, 0. « <a, lim f (w) = f(x), then a necessary and 
n= ce 
sufficient condition that f(a) should have a jinite derivative at 
O ws: the quasi-uniform convergence of the sequence of functions 
yf (®)— af, (y) + DO) 
~ (2,4) = ———_—— — — at the points of every set(a,,y,), 
v& Y 
(Wa, Ya), where OZ w, „<4, OS y, <a, and lim a = 0, lim y= 0. 
k= k— 
4. We observe that in this theorem nothing is supposed about 
the convergence or divergence of the sequence f (0). Let this sequence 
be convergent. Then the sequence of continuous functions f* (2) is 
Yn 
convergent for O<*#<a and converges to f*(#) for « >0, to À 
= lim f (0) for «=0. 
If 70) = lim f (0) then 
it EN aye, RTO ape) liek a 
x=0 
We now consider a set of points «,,#,,..., where lim «= 0. 
k=o 
By «,.V we denote two given positive numbers. Then there exists 
an index NV, > N such that 
OV EEN ee he We AE (6) 
. *, a Ie . . 
Since Sy, = fy) and Sy) is continuous and because of (3), 
there exists a certain number /, such that for every k > &, 
ARE) Ae) Se 
Since at each of the finite number of points where k <%, an 
index Vz; >> N can kj aa such that 
AC mis (er) <é, 
it appears that the ne of functions /" (w) converges quasi- 
uniformly at the points of the considered set. — 
If this sequence of functions converges quasi-uniformly at the 
points of every set of the above-considered kind, then it results from 
an argument as held in § 2 that 
7 (0) = hm En (0). 
n= 
We have therefore the following theorem: 
If a series of functions f («) be given, each of which is continuous 
