In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative fTemplate:'(a) of a function f at a point a:
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 , 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.