Metric differential

In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. With this definition of a derivative, one can generalize Rademacher's theorem to metric space-valued Lipschitz functions.

Discussion

Rademacher's theorem states that a Lipschitz map f : Rⁿ → R^m is differentiable almost everywhere in Rⁿ; 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 Rⁿ that takes values instead in a metric space 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 exists almost everywhere, this does not hold. For example, consider the function f : [0,1] → L¹([0,1]), mapping the unit interval into the space of integrable functions, defined by f(x) = χ_[0,x], this function is Lipschitz (and in fact, an isometry) since, if 0 ≤ x ≤ y≤ 1, then

${\displaystyle |f(x)-f(y)|=\int _{0}^{1}|\chi _{[0,x]}(t)-\chi _{[0,y]}(t)|\,dt=\int _{x}^{y}\,dt=|x-y|,}$

but one can verify that lim_h→0(f(x + h) −  f(x))/h does not converge to an L¹ function for any x in [0,1], 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 Rⁿ which is a function on Rⁿ defined by the limit

${\displaystyle MD(f,z)(x)=\lim _{r\rightarrow 0}{\frac {d_{X}(f(z+rx),f(z))}{r}}}$

whenever the limit exists (here d _X denotes the metric on X).

A theorem due to Bernd Kirchheim[1] states that a Rademacher theorem in terms of metric differentials holds: for almost every z in Rⁿ, MD(f, z) is a seminorm and

${\displaystyle d_{X}(f(x),f(y))-MD(f,z)(x-y)=o(|x-z|+|y-z|).}$

The little-o notation employed here means that, at values very close to z, the function f is approximately an isometry from Rⁿ with respect to the seminorm MD(f, z) into the metric space X.

References

1. Kirchheim, Bernd (1994). "Rectifiable metric spaces: local structure and regularity of the Hausdorff measure". Proc. Am. Math. Soc. 121: 113–124.