Fundamental increment lemma

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In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative f'(a) of a function f at a point a:

The lemma asserts that the existence of this derivative implies the existence of a function such that

for sufficiently small but non-zero h. For a proof, it suffices to define

and verify this meets the requirements.

Differentiability in higher dimensions[edit]

In that the existence of uniquely characterises the number , the fundamental increment lemma can be said to characterise the differentiability of single-variable functions. For this reason, a generalisation of the lemma can be used in the definition of differentiability in multivariable calculus. In particular, suppose f maps some subset of to . Then f is said to be differentiable at a if there is a linear function

and a function

such that

for non-zero h sufficiently close to 0. In this case, M is the unique derivative (or total derivative, to distinguish from the directional and partial derivatives) of f at a. Notably, M is given by the Jacobian matrix of f evaluated at a.

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