653 
the sum of an even number of terms is always zero. For w—=1 
however the series is not convergent. 
6. In this example f(m) converges for «—1 toa limit f(1—0)—0. 
That this need not be the consequence of the quasi-uniform con- 
vergence within the interval OSe <1 is demonstrated by the 
following example: 
Let y=/ (x) be represented by the zig-zag-line A,B,A,B,..., 
MEGA a eee renthienpom istie lf, and — 0} 
B,, B,, B,,... the points with «= },2,;,... and y=1. Now 
choose on the axis of y a countable set of points P,, P,, P;,...> 
everywhere dense in the interval (0,1). Let the function y= S,(w) 
be represented by the following line: from the right to the left first 
the zig-zag A,b,A,b,....AnB,, then the segment of B,A,41 to 
the point of intersection, C,, with the line P= ips last the line 
CP. Evidently S,(e) is continuous in the interval O<x<1. Also 
lim S@) = f(x) for 0< x <1, since from a certain value of n on- 
n— > oo 
wards S,(7) coincides with f(x) for a thus situated point. Hence the 
series S,(x) + {S,(c)—S,(x)} + -..+ {S,(z)—S,—1(z)} +... converges 
in the interval O< x<1 to f(x) and all the terms are continuous 
within O<2<1. This convergence is quasi-uniform. 
In order to make this clear we choose e and N arbitrarily. Since 
the set (P;) is everywhere dense within (0,1), we can now choose 
a finite number of points P,,,Pn,,-.- Pi, between O and 1 of which 
the indices are >> N and which divide the interval into &-+ 1 
segments all < «. Now, let « be an arbitrary value between O and 1. 
If (x) coincides with one of the & values Sn (2), then 
| f (v) — Sn, =O Ge: 
If f(z) does not coincide with any of these values, then 
Sn, (2) = Vp, ded A ey, 
i 
Since 0 < f(r) <1 one of the & indices satisfies 
LF @) — Sn, (@) | Se 
Hereby the quasi-uniform character of the convergence in the 
interval O< «<1 is established. At 0, however, /(#) does not 
assume a limiting value, but oscillates between O and 1. 
UL. 
7. Let flo), fr)... be functions which are continuous within 
the interval a <r<b and let the series 
