Differentiable function/D in R/Linear approximation/Fact/Proof

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Proof

If is differentiable, then we set

Then the only possibility to fulfill the conditions for is

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Because of differentiability, the limit

exists, and its value is . This means that is continuous in .
If and exist with the described properties, then for the relation

holds. Since is continuous in , the limit on the left-hand side, for , exists.