( 62 ) 



have their oscillations below e in the intervals mentioned. On account 

 of the uniform continuity of ƒ we may furthermore have each x 

 included by an interval i' chosen in such a way that for all A g. A' 

 the corresponding <f A have within those intervals oscillations below 

 6 only ; to that end we have but to choose i' in such a way that 

 the oscillations of ƒ remain within the intervals below 7s * A'- If 

 thus i" is the smaller of the two quantities i and i', each x can be 

 included by an interval i" in such a way, that the oscillations of all 

 difference quotients within it remain below s, with which we have 

 proved the uniform continuity of the difference quotients. 



Theorem 3. If there is among the difference quolients of a finite 

 continuous function a uniform continuous fundamental series with 

 indefinitely decreasing ^-increases, there exists a finite continuous 

 differential quotient. 



Let namely <pv{x),(pv{x), .... be the fundamental series of func- 

 tions under consideration, then for any quantity s we can point 

 out a quantity a in such a way that (f^ v ) (x -f h) — <p^ v ) (x) < e for 

 any x, any h<^o and any v. If now the set of all difference quotients 

 were not uniformly continuous, it would have to occur that for 

 a certain A° not belonging to the fundamental series we should have 

 (p AQ (# _j_ h) — <p A o {x) > e. If we now approximate A° by a series 

 a x A', a, A", . . . , where the «'s represent integers, in such a way that 



a p A w <A <K+ l ) A °' ) > then also <PA°( x + h ) — ^a c (- p ) is approxim- 

 ated by (p Ap) {x + h) — <p fp) (a?), which last expression always 



a p x p 



remains < s however large p may become, so that cp A o(x -\- h) — <fi A °(x) 



cannot be > e, so the set of all difference quotients is uniformly 

 continuous, and there is a finite continuous differential quotient. 



Theorem 1 is applied when building up the theory of continuous 

 groups out of the theory of sets, (where one remains independent of 

 Lie's postulates), in a certain region finite and continuous functions 

 of one or more variables occurring there, whose difference quotients 

 are in a certain system of coordinates linear functions of the original 

 functions. l ) As on account of the finiteness of the original functions 

 there cannot be a region within which any quantity could be over- 

 stepped everywhere by one and the same difference quotient, the 



i) Gomp. L. E. J. Brouwer, "Die Theorie der endlichen continuierlichen Gruppen 

 unabhangig von den Axiomen von Lie", Atti del IV<> Gongresso Internazionale 

 dei Materaatici. It is the differentiability in one and the same system of coordinates 

 of all the functions, which express the different infinitesimal transformations of a 

 group, which is proved in this way. 



