# 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:

${\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}.}$

The lemma asserts that the existence of this derivative implies the existence of a function ${\displaystyle \varphi }$ such that

${\displaystyle \lim _{h\to 0}\varphi (h)=0\qquad {\text{and}}\qquad f(a+h)=f(a)+f'(a)h+\varphi (h)h}$

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

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

and verify this ${\displaystyle \varphi }$ meets the requirements.

## Differentiability in higher dimensions

In that the existence of ${\displaystyle \varphi }$ uniquely characterises the number ${\displaystyle f'(a)}$, 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 ${\displaystyle \mathbb {R} ^{n}}$ to ${\displaystyle \mathbb {R} }$. Then f is said to be differentiable at a if there is a linear function

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

and a function

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

such that

${\displaystyle \lim _{{\mathbf {h}}\to 0}\Phi ({\mathbf {h}})=0\qquad {\text{and}}\qquad f({\mathbf {a}}+{\mathbf {h}})=f({\mathbf {a}})+M({\mathbf {h}})+\Phi ({\mathbf {h}})\cdot \Vert {\mathbf {h}}\Vert }$

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.