420 
From (2) and (3) it follows that Pe, vl Sé for “<d, UK J 
and from this, since /im x, —=0 and lim y = 0, 
CS == D 
lim P(x ,y)=90. a A tink Tabel bet Bn 
k= 
From (1) and (4) it follows that M == m, hence lim f(x) exists 
Tal 
and is finite, unless m == Jo or M= — oo. 
If however m= + o, then at the points of every set x, 
where «, > 0 and fim x, =0, the lim f(x) is + oo. 
k= 
k= 
To every «, of such a set a y, >0 can be found such that 
JW) > Fe) +k and oR — Oee then Ge (ey) =— @, which 
is incompatible with the formula (4). Similarly it appears that M 
cannot possibly be — oo. Collecting our results we have the following 
theorem : 
If a sequence of functions f (x) be given each of which is conti- 
nuous for OSe <a, and if lim f (x) = f(2) for Oe Sa, then a 
n= 0 
necessary and sufficient condition that f(x) should tend to a finite 
limit at O vs: the quasi-uniform convergence of the sequence of functions 
P, (x,y) = f (@) —f(y) at the Mounts Of every sch IE Gen een 
where Ow, Sa, OS y, Sa and lina, =O, lim ye = 0. 
k= om k= 
Il. 
3. Let f(2), f,(@),... be continuous for O<x<a with finite 
derivatives f (0) at 0. Let the series be convergent for 0 Sa Sa. 
sha A AET : 
The funetions J. (@) = for « >0 and f (O) for «=O are 
46 
in consequence of the suppositions continuous atO<w<a; atOSa<a 
the sequence of functions f (w) converges to the limiting function 
; t)—f (©, mals : 
TED Ja) JO) , where /f(#) = lm Ff (2). 
x n=O 
If f(v) has a finite derivative at 0, then /*(x) tends to a finite 
limit for v—0, and vice versa. 
Hence, by applying the theorem demonstrated in I, we find, if 
we take into consideration that 
f (e) —af (y) + ef, ©) 
avy i 
ol 
Of (= 
