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:
“An absolute can only be given in an intuition, while all the rest has to do with analysis. We call intuition here the sympathy by which one is transported into the interior of an object in order to coincide with what there is unique and consequently inexpressible in it. Analysis, on the contrary, is the operation which reduces the object to elements already known.”
—Henri Bergson (1859–1941)
“Every society consists of men in the process of developing from children into parents. To assure continuity of tradition, society must early prepare for parenthood in its children; and it must take care of the unavoidable remnants of infantility in its adults. This is a large order, especially since a society needs many beings who can follow, a few who can lead, and some who can do both, alternately or in different areas of life.”
—Erik H. Erikson (1904–1994)
“Mark the babe
Not long accustomed to this breathing world;
One that hath barely learned to shape a smile,
Though yet irrational of soul, to grasp
With tiny finger—to let fall a tear;
And, as the heavy cloud of sleep dissolves,
To stretch his limbs, bemocking, as might seem,
The outward functions of intelligent man.”
—William Wordsworth (1770–1850)