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

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

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 ${\textstyle h}$. 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.

The lemma says, at least when ${\displaystyle h}$ is sufficiently close to zero, that the difference quotient

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

can be written as the derivative f' plus an error term ${\displaystyle \varphi (h)}$ that vanishes at ${\displaystyle h=0}$.

I.e. one has,

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

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.

We can write the above equation in terms of the partial derivatives ${\displaystyle {\frac {\partial f}{\partial x_{i))))$ as

${\displaystyle f(\mathbf {a} +\mathbf {h} )-f(\mathbf {a} )=\displaystyle \sum _{i=1}^{n}{\frac {\partial f(a)}{\partial x_{i))}+\Phi (\mathbf {h} )\cdot \Vert \mathbf {h} \Vert }$