Kneser's theorem (combinatorics)

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:

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

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 fTemplate:'(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

• Template:Cite web
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