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:

f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}.

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

\lim_{h \to 0} \varphi(h) = 0 \qquad \text{and} \qquad f(a+h) = f(a) + f'(a)h + \varphi(h)h

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

\varphi(h) = \frac{f(a+h) - f(a)}{h} - f'(a)

and verify this \varphi meets the requirements.

Differentiability in higher dimensions[edit]

In that the existence of \varphi 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 \mathbb{R}^n to \mathbb{R}. Then f is said to be differentiable at a if there is a linear function

M: \mathbb{R}^n \to \mathbb{R}

and a function

\Phi: D \to \mathbb{R}, \qquad D \subseteq \mathbb{R}^n \smallsetminus \{ \bold{0} \},

such that

\lim_{\bold{h} \to 0} \Phi(\bold{h}) = 0 \qquad \text{and} \qquad f(\bold{a}+\bold{h}) = f(\bold{a}) + M(\bold{h}) + \Phi(\bold{h}) \cdot \Vert\bold{h}\Vert

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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