Metric differential
# 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 : $R^n$ → $R^m$ is differentiable almost everywhere 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 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] → $L^1([0,1]),$ 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

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

but one can verify that $lim_{h→0}(f(x$ + h) − f(x))/h does not converge to an $L^1$ 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 $f:R^n$ → 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

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

whenever the limit exists (here $d_X$ denotes the metric on X).

A theorem due to Bernd Kirchheim states that a Rademacher theorem in terms of metric differentials holds: for almost every z in $R^n,$ MD(f, z) is a seminorm and

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

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

