( 61 ) 



So if' we regard for any definite x all difference quotients the 

 «-increases of which are equal to proper fractions of A, then the amount 

 t a (#) of their region of oscillation is smaller than 2s A (x). The same 

 holds for the region of oscillation of all difference quotients for 

 definite x with «-increases smaller than A, because these can be 

 approximated by the preceding on account of the continuity of ƒ. 



So if we allow A to decrease indefinitely, then also t a («) decreases 

 indefinitely ; as furthermore when A becomes smaller, each following 

 region of oscillation is a part of the preceding, the limit of the 

 region of oscillation is for each x a single definite value, to which 

 uniform convergence takes place, which is the limit of the difference 

 quotients, the differential quotient. 



That this (forward) differential quotient cannot show any disconti- 

 nuities, is evident as follows : If there were a discontinuity, then 

 there would be a quantity a which could be overstepped there for any 

 interval by the oscillations of the differential quotient; but if the 

 value of the differential quotient differs in two points more than a, 

 then there is also a difference quotient the values of which in 

 those two points differ more than o-, so there would be for each 

 interval, which contains the indicated discontinuity, a difference 

 quotient with a region of oscillation larger than a, i. o. w. the functional 

 set of the difference quotients would not be uniformly continuous. 



Out of the continuity of the forward differential quotient follows 

 at the same time that the forward and the backward differential 

 quotient are equal. 



Analogously it is evident that also a point at infinity in the differential 

 quotient would disturb the uniform continuity of the difference 

 quotients; in this is at the same time included the limitedness of 

 the difference quotients, for they would otherwise on account of the 

 finiteness of ƒ be able to tend to infinity only for indefinitely 

 decreasing «-increase, but that would furnish an infinity point in the 

 differential quotient. 



Theorem 2. Of a function with finite continuous differential 

 quotient the difference quotients are uniformly continuous. 



Let namely e be a definite quantity, to be taken as small as one 

 likes. Now we may have each x included by an interval i in 

 such a way, that the oscillations of the differential quotient within 

 each of those intervals remain smaller than V» e. On account of the 

 uniform convergence, evident from the formula <p A (x) =f(x-\- # A), 

 a A' can be pointed out in such a way that all y* for which A < A' 

 differ from the differential quotient less than l / t € for any x, thus 



