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In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus, differentiation and integration, expressed by the fundamental theorem of calculus in the framework of Riemann integration. Such generalizations are often formulated in terms of Lebesgue integration. For real-valued functions on the real line two interrelated notions appear: ''absolute continuity of functions'' and ''absolute continuity of measures.'' These two notions are generalized in different directions. The usual derivative of a function is related to the ''Radon–Nikodym derivative'', or ''density'', of a measure. We have the following chains of inclusions for functions over a compact subset of the real line: : absolutely continuous ⊆ uniformly continuous ⊆ continuous and: : Continuously differentiable ⊆ Lipschitz continuous ⊆ absolutely continuous ⊆ bounded variation ⊆ differentiable almost everywhere ==Absolute continuity of functions== A continuous function fails to be absolutely continuous if it fails to be uniformly continuous – examples are tan(''x'') over (), exp(''x'') over the entire real line, and sin(1/''x'') over (0, 1 )). But there is another way for a continuous function ''f'' to fail to be absolutely continuous – if it is differentiable almost everywhere in an interval and its derivative ''f'' ′ is Lebesgue integrable, but the integral of ''f'' ′ differs from the increment of ''f''. For example, this happens for the Cantor function. 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「Absolute continuity」の詳細全文を読む スポンサード リンク
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