Foundation of Paralogical Nonstandard Analysis and its Application to Some Famous Problems of Trigonometrical and Orthogonal Series

This is an article about foundation of paralogical nonstandard analysis and its applications to the continuous function without a derivative presented by absolutely convergent trigonometrical series and another famous problems of trigonometrical and orthogonal series. In part 1 of the present work, using the methods of paralogical nonstandard analysis, we shall obtain a general criterion for that there is no function which would be the almost everywhere finite derivative function for the following continuous function