concept of mil form convergence 151 



A3: Uniform convergence at a point. The series is 

 said to be uniformly convergent at the point x = f {or for x = ^) 

 if to every positive e correspond a positive S (^, e) and an 

 »o(|, €, B) = nQ(^, e) such that (A) is true for n ^n^{^, e) and for 

 ^-S(^,e)^x^^+S(^,e). 



4. Before proceeding further it will be well to make a few 

 remarks concerning these definitions and their relations to one 

 another. 



The idea of uniform convergence in the neighbourhood of a 

 particular point (Definition A 2) is substantially that defined by 

 Seidel in 1848*. It is clear, however, that definitions A 1 and 

 A 2 were both familiar to Weierstrass as early as 1841 or 1842f. 

 It is obvious that a series uniformly convergent throughout an 

 interval is uniformly convergent in the neighbourhood of every 

 point of the interval. The converse theorem is important and by 

 no means obvious, and was first proved by Weierstrass | in a memoir 

 published in 1880. This theorem would now be proved by a 

 simple application of the ' Heine-Borel Theorem ', and is a par- 

 ticular case of a theorem which will be referred to in a moment. 



Definition A3 appears first, in the form in which I state it, in 

 a paper of W. H. Young published in 1903§; but the idea is 

 present in an earlier paper of Osgood ||. The essential difference 

 between definitions A 2 and A 3 is that in the latter S is chosen 

 after e and is a function of ^ and e, while in the former it is chosen 

 before e and is a function of f alone. In each case n^ is a function 

 of two independent variables, ^ and e. It is plain that uniform 

 convergence in the neighbourhood of ^ involves uniform conver- 

 gence at ^, and at (and indeed in the neighbourhood of) all points 

 sufficiently near to ^. But uniform convergence at ^ does not 

 involve uniform convergence in the neighbourhood of |. 



It is important, however, to observe that uniforni convergence 

 at every point of an interval involves uniform convergence throughout 

 tJie interval. This important theorem is proved very simply by 



* ' Note iiber eine Eigensehaft der Reihen, welche discontinuirliche Functionen 

 darstellen', Munchener Ahliandlungen, vol. 7, 1848, pp. 381-394. This memoir has 

 been reprinted in Ostwald's Klassiker der e.vakten Wisscmchaften, no. IK!. The 

 reference tliere given to vol. 5, 1847, is incorrect. 



(■ For detailed references bearing on this and similar historical points, see 

 Pringsheim's article already qnoted. 



X See the memoir 'Zur Functionenlehre ' {Ahliandlungen aus der Funktionen- 

 lehre, pp. 69-104 (pp. 71-72)). 



§ 'On non-uniform convergence and term-by-term integration of series', Proc. 

 London Math. Soc, ser. 2, vol. 1, pp. 89-102. 



II 'Non-uniform convergence and the integration of series', American Journal of 

 Math., vol. 19, 1897, pp. 155-190. See Prof. Young's remarks on this point at the 

 beginning of his later paper ' On uniform and non-uniform convergence of a series 

 of continuous functions and the distinction of right and left ', Proc. London Math. 

 Soc, ser. 2, vol. 6, 1907, pp. 29-51. 



