Epsilon-complexity of continuous functions

A formal definition of ǫ-complexity of an individual continuous function defined on a unit cube is proposed. This definition is consistent with the Kolmogorov’s idea of the complexity of an object. A definition of ǫ-complexity for a class of continuous functions with a given modulus of continuity is also proposed. Additionally, an explicit formula for the ǫ-complexity of a functional class is obtained. As a consequence, the paper finds that the ǫcomplexity for the Hölder class of functions can be characterized by a pair of real numbers. Based on these results the papers formulates a conjecture concerning the ǫ-complexity of an individual function from the Hölder class. We also propose a conjecture about characterization of ǫ-complexity of a function from the Hölder class given on a discrete grid.