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:
“Then the justice,
In fair round belly with good capon lined,
With eyes severe and beard of formal cut,
Full of wise saws and modern instances;
And so he plays his part.”
—William Shakespeare (1564–1616)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)