( 64) 



the difference for «r-increase A 2 of the (first) differential quotient can 

 therefore not differ more than b' Ai (x) from the just deduced limiting 

 function which is thus differential quotient of the (first) differential 

 quotient i.e. second differential quotient. 



Theorem 5. If a function possesses a finite continuous second 

 differential quotient, then the system of the first and second difference 

 quotients is uniformly continuous. 



To find namely an interval size i" which keeps the oscillations 

 of all second difference quotients everywhere <[ e, we first take 

 the interval size i, which keeps the oscillations of the second 

 differential quotient everywhere <C è 8 ; then a A\ and a A', in 

 such a way, that all yA,A 3 , for which h 1 <C A\ and A 2 < A' s , 

 differ along the whole course less than 7 4 8 from the second diffe- 

 rential quotient l ) ; finally an interval size i' which keeps the oscil- 

 lations of the function ƒ everywhere < l /« e A', A',. For i" we take 

 the smaller of the two quantities i and i' . 



Theorem 6. If there is among the second difference quotients of a 

 finite continuous function with finite continuous differential quotient 

 a uniformly continuous fundamental series, in which the two x- 

 increases decrease indefinitely, then there exists a finite continuous 

 second differential quotient. 



Let namely <p » - (x) , <p » n(x)... be the indicated fundamental 



series, then for any quantity e a quantity o can be pointed out 

 in such a way, that <p ^ ( y ) ( vj {x + /*) — <p r,) ( v ) {x) < e for any x, 



12 12 



any h<^G and any v. If now the set of all second difference 

 quotients were not uniformly continuous, then it would be possible 

 for a certain A^ and A 3 ° not belonging to the fundamental series, 

 that <poo (x -\- h) — <p A o A o (x)^> e. Let us now approximate Aj 



by means of a series «jA,' , « a A/' , . . . . and A 3 ° by means of a 

 series &A,' , iS,A," , . . ., where the «'s and 0's represent integers, in 

 such a way that 



a p h[ p) < A° < (a p + 1) A[ /0 and P p h[ p) <^< (ft, + 1) A^ } , 



!) The uniform convergence of all difference quotients is evident from that of the 

 difference quotients, for which Aj = A 2 (out of these the other can be approxi- 

 mated in the manner indicated in the proof of theorem 6); the latter is evident 

 by developing the terms of f{x + 2 A) — 2 f {x + A) + f (x) according to Taylor's 

 series, in which we make the second differential quotient form the restterm ; the 

 terms preceding this restterm then destroy each other. 



