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 → R is differentiable almost everywhere in R; 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 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"([0,1]), mapping the unit interval into the space of integrable functions, defined by "f"("x") = "χ", this function is Lipschitz (and in fact, an isometry) since, if 0 ≤ "x" ≤ "y"≤ 1, then but one can verify that lim("f"("x" + "h") −  "f"("x"))/"h" does  not converge to an "L" 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 → "X" is the metric differential of "f" at a point "z" in R which is a function on R defined by the  limit whenever the limit exists (here "d" 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, MD("f", "z") is a seminorm and The little-o notation employed here means that, at values very close to "z", the function "f" is approximately an isometry from R with respect to the seminorm MD("f", "z") into the metric space "X".