In single-variable
differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the
derivative of a
function at a point :
The lemma asserts that the existence of this derivative implies the existence of a function such that
for sufficiently small but non-zero . For a proof, it suffices to define
and verify this meets the requirements.
The lemma says, at least when is sufficiently close to zero, that the difference quotient
can be written as the derivative f' plus an error term that vanishes at .
I.e. one has,
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