Fundamental increment lemma - Alchetron, the free social encyclopedia

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:

f ′ ( a ) = lim h → 0 f ( a + h ) − f ( a ) h .

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

lim h → 0 φ ( h ) = 0 and f ( a + h ) = f ( a ) + f ′ ( a ) h + φ ( h ) h

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

φ ( h ) = f ( a + h ) − f ( a ) h − f ′ ( a )

and verify this φ meets the requirements.

Differentiability in higher dimensions

In that the existence of φ uniquely characterises the number f ′ ( a ) , 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 R n to R . Then f is said to be differentiable at a if there is a linear function

M : R n → R

and a function

Φ : D → R , D ⊆ R n ∖ { 0 } ,

such that

lim h → 0 Φ ( h ) = 0 and f ( a + h ) = f ( a ) + M ( h ) + Φ ( h ) ⋅ ∥ h ∥

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.

References

Fundamental increment lemma Wikipedia

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