Continuity module

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 ${\ displaystyle f}$

{\ displaystyle \ epsilon _ {f} (\ delta): = \ sup \ left \ {d (f (x), f (y)) \ colon d (x, y) \ leq \ delta \ right \}}

defined function . ${\ displaystyle \ epsilon _ {f} \ colon \ left [0, \ infty \ right) \ to \ mathbb {R}}$

The modulus of continuity increases monotonically with . ${\ displaystyle \ epsilon _ {f} (0) = 0}$
The continuity module is subadditive : for everyone . ${\ displaystyle \ epsilon _ {f} (\ delta _ {1} + \ delta _ {2}) \ leq \ epsilon _ {f} (\ delta _ {1}) + \ epsilon _ {f} (\ delta _ {2})}$ ${\ displaystyle \ delta _ {1}, \ delta _ {2} \ geq 0}$
${\ displaystyle f}$ is uniformly continuous if and only if . ${\ displaystyle \ textstyle \ lim _ {\ delta \ to 0} \ epsilon _ {f} (\ delta) = 0}$
${\ displaystyle f}$ is Lipschitz continuous with Lipschitz constant if and only if holds for all . ${\ displaystyle L}$ ${\ displaystyle \ epsilon _ {f} (\ delta) \ leq L \ delta}$ ${\ displaystyle \ delta}$
${\ displaystyle f}$ is Hölder continuous with Hölder exponent if and only if there is a constant with for all . ${\ displaystyle \ alpha}$ ${\ displaystyle C}$ ${\ displaystyle \ epsilon _ {f} (\ delta) \ leq C \ delta ^ {\ alpha}}$ ${\ displaystyle \ delta}$