Fundamental increment lemma

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

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.

See also

• Generalizations of the derivative