Absolute Continuity of Functions
It may happen that a continuous function f is differentiable almost everywhere on, its derivative f ′ is Lebesgue integrable, and nevertheless the integral of f ′ differs from the increment of f. For example, this happens for the Cantor function, which means that this function is not absolutely continuous. Absolute continuity of functions is a smoothness property which is stricter than continuity and uniform continuity.
Read more about this topic: Absolute Continuity
Famous quotes containing the words absolute, continuity and/or functions:
“As liberty of thought is absolute, so is liberty of speech, which is “inseparable” from the liberty of thought. Liberty of speech, moreover, is essential not only for its own sake but for the sake of truth, which requires absolute liberty for the utterance of unpopular and even demonstrably false opinions.”
—Gertrude Himmelfarb (b. 1922)
“The dialectic between change and continuity is a painful but deeply instructive one, in personal life as in the life of a people. To “see the light” too often has meant rejecting the treasures found in darkness.”
—Adrienne Rich (b. 1929)
“In today’s world parents find themselves at the mercy of a society which imposes pressures and priorities that allow neither time nor place for meaningful activities and relations between children and adults, which downgrade the role of parents and the functions of parenthood, and which prevent the parent from doing things he wants to do as a guide, friend, and companion to his children.”
—Urie Bronfenbrenner (b. 1917)