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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 Rademarcher'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'' : () → ''L''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''h''→0(''f''(''x'' + ''h'') − ''f''(''x''))/''h'' does not converge to an ''L''1 function for any ''x'' in (), 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Metric differential」の詳細全文を読む スポンサード リンク
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