mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, a metric differential is a generalization of a

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

for a Lipschitz continuous function defined on a

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

and taking values in an arbitrary

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...

. With this definition of a derivative, one can generalize

Rademacher's theorem In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If is an open subset of and is Lipschitz continuous, then is differentiable almost everywhere in ; that is, the points in at which is ''not'' d ...

to metric space-valued Lipschitz functions.

Discussion

states that a Lipschitz map ''f'' : R^''n'' → R^''m'' is differentiable

almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...

in R^''n''; in other words, for almost every ''x'', ''f'' is approximately linear in any sufficiently small range of ''x''. If ''f'' is a function from a Euclidean space R^''n'' that takes values instead in a

''X'', it doesn't immediately make sense to talk about differentiability since ''X'' has no linear structure a priori. Even if you assume that ''X'' is a Banach space and ask whether a

Fréchet derivative In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued ...

exists almost everywhere, this does not hold. For example, consider the function ''f'' : ,1nbsp;→ ''L''¹( ,1, mapping the unit interval into the space of integrable functions, defined by ''f''(''x'') = ''χ''_{,''x''/sub>, this function is Lipschitz (and in fact, an isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
) since, if 0 ≤ ''x'' ≤ ''y''≤ 1, then

: $, f(x)-f(y), =\int_0^1 , \chi_(t)-\chi_(t), \,dt = \int_x^y \, dt = , x-y, ,$

but one can verify that lim_''h''→0(''f''(''x'' + ''h'') − ''f''(''x''))/''h'' does not converge to an ''L''¹ function for any ''x'' in ,1 so it is not differentiable anywhere.

However, if you look at Rademacher's theorem as a statement about how a Lipschitz function stabilizes as you zoom in on almost every point, then such a theorem exists but is stated in terms of the metric properties of ''f'' instead of its linear properties.

Definition and existence of the metric differential

A substitute for a derivative of ''f'':Rⁿ → ''X'' is the metric differential of ''f'' at a point ''z'' in R^''n'' which is a function on R^''n'' defined by the limit

: $MD(f,z)(x)=\lim_ \frac$
whenever the limit exists (here ''d''_''X'' denotes the metric on ''X'').

A theorem due to Bernd Kirchheim states that a Rademacher theorem in terms of metric differentials holds: for almost every ''z'' in R^''n'', MD(''f'', ''z'') is a seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
and

: $d_X(f(x),f(y)) - MD(f,z)(x-y) = o(, x-z, +, y-z, ).$

The little-o notation

Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
employed here means that, at values very close to ''z'', the function ''f'' is approximately an isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
from R^''n'' with respect to the seminorm MD(''f'', ''z'') into the metric space ''X''.

References

{{Reflist

Lipschitz maps
Mathematical analysis}