The continuity module is a term from the field of mathematical analysis . It was introduced by Henri Lebesgue in 1910 and is used, among other things, in approximation theory , where it is used to establish a connection between the smoothness of a function and the approximation speed when approximating with polynomials .
definition
The continuity module of a function is through
defined function .
properties
- The modulus of continuity increases monotonically with .
- The continuity module is subadditive : for everyone .
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is uniformly continuous if and only if .
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is Lipschitz continuous with Lipschitz constant if and only if holds for all .
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is Hölder continuous with Hölder exponent if and only if there is a constant with for all .
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