Fundamental increment lemma
The lemma asserts that the existence of this derivative implies the existence of a function such that
for sufficiently small but non-zero h. For a proof, it suffices to define
and verify this meets the requirements.
Differentiability in higher dimensions
In that the existence of 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 to . Then f is said to be differentiable at a if there is a linear function
and a function
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.