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 , 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.
- Talman, Louis (2007-09-12). "Differentiability for Multivariable Functions" (PDF). Retrieved 2012-06-28.
- Stewart, James (2008). Calculus (7th ed.). Cengage Learning. p. 942. ISBN 0538498846.