Real functions/Differentiability/Linear approximation/Section

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We discuss a property which is equivalent with differentiability, the existence of a linear approximation. This formulation is important in many respects: It allows giving quite simple proofs of the rules for differentiable functions, one can use it to reduce differentiability to the continuity of an error function, it yields a model for approximation with polynomials of higher degree (quadratic approximation, Taylor expansion), and it allows a direct generalization to the higher-dimensional situation.


Theorem

Let be a subset, a point, and

a function. Then is differentiable in if and only if there exists some and a function

such that is continuous in , , and such that

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.


The affine-linear function

is called the affine-linear approximation. The constant function given by the value can be considered as the constant approximation.


Corollary

Let be a subset, a point, and

a function. Then is also continuous in .

Proof  

This follows immediately from