Differentiable function/D in R/Rules/Fact/Proof

Proof

(1). We write and respectively with the objects which were formulated in fact, that is

and

Summing up yields

Here, the sum is again continuous in , with value .
(2). We start again with

and

and multiply both equations. This yields

Due to fact for limits, the expression consisting of the last six summands is a continuous function, with value for .
(3) follows from (2), since a constant function is differentiable with derivative .
(4). We have

Since is continuous in , due to fact, the left-hand factor converges for to , and because of the differentiability of in , the right-hand factor converges to .
(5) follows from (2) and (4).