Fundamental increment lemma

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

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

The lemma asserts that the existence of this derivative implies the existence of a function $\varphi$ such that

$\lim _{{h\to 0}}\varphi (h)=0\qquad {\text{and}}\qquad f(a+h)=f(a)+f'(a)h+\varphi (h)h$

for sufficiently small but non-zero h. For a proof, it suffices to define

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

and verify this $\varphi$ meets the requirements.

Differentiability in higher dimensions

In that the existence of $\varphi$ 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 $\mathbb {R} ^{n}$ to $\mathbb {R}$. Then f is said to be differentiable at a if there is a linear function

$M:{\mathbb {R}}^{n}\to {\mathbb {R}}$

and a function

$\Phi :D\to {\mathbb {R}},\qquad D\subseteq {\mathbb {R}}^{n}\smallsetminus \{{\mathbf {0}}\},$

such that

$\lim _{{{\mathbf {h}}\to 0}}\Phi ({\mathbf {h}})=0\qquad {\text{and}}\qquad f({\mathbf {a}}+{\mathbf {h}})=f({\mathbf {a}})+M({\mathbf {h}})+\Phi ({\mathbf {h}})\cdot \Vert {\mathbf {h}}\Vert$

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.

See also

• Generalizations of the derivative

References

• 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.