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:
“José’s my first non-rat romance. Not that he’s my idea of the absolute finito. He’s too prim and cautious to be my absolute ideal, Now, if I could choose from anybody alive, I wouldn’t pick José. Nehru, maybe, or Albert Schweitzer. Or Leonard Bernstein.”
—George Axelrod (b. 1922)
“There is never a beginning, there is never an end, to the inexplicable continuity of this web of God, but always circular power returning into itself.”
—Ralph Waldo Emerson (1803–1882)
“The English masses are lovable: they are kind, decent, tolerant, practical and not stupid. The tragedy is that there are too many of them, and that they are aimless, having outgrown the servile functions for which they were encouraged to multiply. One day these huge crowds will have to seize power because there will be nothing else for them to do, and yet they neither demand power nor are ready to make use of it; they will learn only to be bored in a new way.”
—Cyril Connolly (1903–1974)