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.
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