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:

$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}.$

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

$\underset{h\to 0}{\lim }\phi (h)=0\text{and}f(a+h)=f(a)+f^{\prime }(a)h+\phi (h)h$

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

$\phi (h)=\frac{f(a+h)-f(a)}{h}-f^{\prime }(a)$

and verify this $\phi$ meets the requirements.

Differentiability in higher dimensions

In that the existence of $\phi$ 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

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

such that

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

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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