( 66 ) 



quotient, then the system of the first, second, up to the n th difference 

 quotients is uniformly continuous 1 ). 



Theorem 10. If n lf n at n t . ... is an infinite series of increasing 

 integers and if of a finite continuous function the systems of the 

 n st , of the rc, nd .... difference quotients are uniformly continuous, 

 then the function has all differential quotients and these are all finite 

 and continuous. 



Theorem 11. A finite continuous function of several variables, among 

 whose difference quotients of the n th order there is for each kind a 

 uniformly continuous fundamental series in which the increases 

 of the independent variables decrease indefinitely, possesses all 

 differential quotients up to the n th order; these finite and con- 

 tinuous differential quotients are first each other's differential quotients 

 in the manner expressed by their form, where the order of succession 

 of the differentiations proves to be irrelevant, and then each diffe- 

 rential quotient is the only limit of the corresponding difference quo- 

 tients for indefinitely decreasing increases of the independent vari- 

 ables, to which limit a uniform convergence takes place. 



Theorem 12. If a function of several variables possesses all kinds of 

 n th differential quotients and if these are finite and continuous, 

 then the system of the 1 st , 2 nd up to the n lh difference quotients is 

 uniformly continuous.") 



Finally the observation, that what was treated here leads in- 

 finite differentiability back to continuity in a more extensive sense, 

 and in this way may somewhat explain, that for so long all finite 

 continuous functions were supposed to be infinitely differentiable, and 

 may somewhat justify that so many wish to limit themselves in 

 natural science to infinitely differentiable functions. 



i) To prove the uniform convergence of the w th difference quotients for equal 

 independent re-increases we break off just as for theorem 5 the Taylor development 

 at the n th term and we apply the formula: 



l^in-l)" +( 2 )(n-2f = 



(w and a integers ; a < n). 



2) To prove the uniform convergence of the nrt> difference quotients with equal 

 independent increases A (these & indefinitely decreasing) we develop the elements 

 of such a difference quotient according to Taylor's series, in which we make the 

 nth differential quotients form the restterms. The terms preceding the restterms 

 then fall out and of the restterms one kind converges uniformly to the differential 

 quotient corresponding to the difference quotient considered and the other converge 

 uniformly to zero. 



