Formal Definition
Formally, a modulus of continuity is any real-extended valued function
vanishing at 0 and continuous at 0, that is
Moduli of continuity are mainly used to give a quantitative account both of the continuity at a point, and of the uniform continuity, for functions between metric spaces, according to the following definitions.
A function admits as (local) modulus of continuity at the point if and only if,
Also, admits as (global) modulus of continuity if and only if,
One equivalently says that is a modulus of continuity (resp., at ) for, or shortly, is -continuous (resp., at ). Here, we mainly treat the global notion.
Read more about this topic: Modulus Of Continuity
Famous quotes containing the words formal and/or definition:
“The bed is now as public as the dinner table and governed by the same rules of formal confrontation.”
—Angela Carter (1940–1992)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)