934 



second limit is there at least and the first at most ( — ) times as 



«mail as in the point «j. From this it at once follows that the first 

 limit cannot be equal to zero in anj such point « nor can 

 the total limit. Further we have for two arbitrary points « and 

 «' of the interval (0, a^) that the ratio of the values which the first 



limit assumes there, lies between f — j and 1, and thus approaches 



to 1 as «' approaches to «. In other words the first limit and thus 

 the total limit too, which is equal to it, is a continuous function 

 of a in the interval (0, «J. 



From the hypothesis that a^it^) [s finite and (liferent from zero 

 we have thus inferred that a (a) is continuous in the interval (0, «J 

 and also different from zero. But if a («,) differs from zero the'same 

 holds for any value of a («) corresponding to a point « of the 

 interval («i, ^). Thus we infer that the function aia) is continuous 

 and dijj event from zero in any sub-interval of (0, A) which has the 

 left-hand end-point in common iinth (0, A), and thus shortly speaking 

 in (0, A). The limiting values (93) are thus in the internal points of 

 the interval (0, A) either all equal to zero or all different from zero, 

 but also in the latter case they form a function of u which is 

 continuous within that interval. This is the result required. 



As regards the end points of the interval (0, A) the foregoing 

 reasoning does not inform us of anything. We may with a view 

 to these points distinguish the following cases, which all are possible 

 as it appears from the examples added. 



1^^ . The function a («) is continuous in both endpoints, that is to 

 say continuous on the right at a=zO, and continuous o7i the left 

 at a=z A. 



Examples : a,„ {or) = 1 + x'^\ The interval (0, R) where the functions 

 am («) can be expanded into a power-series of a, is here the interval (0, oo) ; 

 the maximum value A of the «-values for which a («) is finite is 

 equal to unity. The function a ^«) is in the closed interval (0,1) 

 equal to the constant value 1. As another example w^e may quote 

 am ix) = X'" -f- ^^'"'j foi* which also Z? = oo, A = 1; here a («) is in 

 the closed interval (0,1) equal to x. More generally we may take 



a,„ (a-) = 3/'« (1 -f .?;'«'), 



where y^f{x) is a function of x with a radius of convergence 

 greater than unity, and is identical with its natural majorant. Here 

 we again have ^4^1 and /? = the just-mentioned radius of conver- 

 gence of y ; the function a {a) is in the closed interval (0,1) identical 



