concept of uniform convergence 153 



relation to Stokes's memoir. For the necessary and sufficient con- 

 dition that s (x) should he continuous for x=^ is that the series 

 should be quasi-uniformly convergent for x — ^. This theorem is 

 in substance due to Dini*. I give the proof, as it is essential for 

 the criticism of Stokes's memoir. 



(1) The condition is siificient. For 



I s {x) - s (I) I ^ { Sn {x) - Sn (|) ' + | r„ {x) j + | r^ (f) |. 



Choose e, N, S (^, e, N), and n = ??o {^, e, N) as in definition B 3. Then 

 [ r„ {x) I < e for ^—Z^x^^ + h. Now that n is fixed we can choose 

 Si less than 8 and such that [ s,i {x) — «» (^) j < e for ^ — Si ^ a" ^ ^ + Sj. 

 And thus 



|s(.«)-s(f)i<3e 



for ^ — §1 ^ .« ^ ^ + Si , so that 5 {x) is continuous for a? = f . 



It is plain that this argument proves, a fortiori, that A 2, A 3, 

 and B 2 all furnish sufficient conditions for continuity at a point, 

 and A 1 and B 1 sufficient conditions for continuity throughout an 

 interval. 



(2) The condition is necessary. For 



I rn {x) \^\S {x) - S (^) I + ! Vn (f ) | + ! S„ {x) - 5„ (|) |. 



Suppose that e and N are given. Then we can choose S (^, e) 

 so that \s{x) — s{^)\<e for f — S ^ « ^ ^ + S, and n^ (|, e, iV ) so that 

 Vq > N and j r^^ (|) | < e. And, when n^ has thus been fixed, we can 

 choose S] (^, e, n^) = Sj (^, e, N) so that Si < S and 1 6'„^ {x) — Sn^ (f ) | < e 

 for I — Si ^ .^■ ^ I + Si . Thus | r,i (^) j < 3e for n = no> N and 

 ^ — Bi^X'^^ + 8i, so that the series is quasi-uniformly convergent 

 for x=^. 



6. If a series is uniformly convergent at every point ^ of an 

 interval, it is (as we saw in § 4) uniformly convergent throughout 

 the interval : definition A 3 (and a fortiori definition A 2) passes 

 over, in virtue of the Heine-Borel Theorem, into definition A 1. 

 It is important to observe that this relation does not hold between 

 B 3 (or B 2) and B 1 : a series quasi-uniformly convergent at 

 every point of an interval (or in the neighbourhood of every such 

 point) is not necessarily quasi-uniformly convergent throughout 

 the interval. We can apply the Heine-Borel Theorem in the 

 manner indicated in the first sentences of the footnote * to p. 152 ; 

 but the last stage of the argument, in which every one of a finite 

 number of difterent integers is replaced by the largest of them, 

 fails. What we obtain is the necessary and sufficient condition that 

 s {x) shoidd he continuous throughout the interval ; and this is not 



^' Foiulaiiii')iti..., p. 107 ((jerinan translation, GruiuUa(ii'ii...,p\). 143-145). 



