Differentiable function/D open in K/Rules/Fact/Proof

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


Summing up yields

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


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 differentiablity of in the right hand factor converges to .
(5) follows froms (2) and (4).