ONE CLASS OF CONTINUOUS FUNCTIONS WITH COMPLICATED LOCAL PROPERTIES RELATED

摘要

In the paper, we construct and study the class of continuous on 0, 1 functions with continuum set of peculiarities (singular, nowhere monotonic, and non-differentiable functions are among them). The representative of this class is the function y = f(x) defined by the Engel representation of argument: where gn = gn(x) ISIN; {0,1, 2, ...}, and convergent real series E INFIN; n=0 un= u0 + u1+ ... + un + rn = 1, un< 1, 0 < rn< 1, by the following equality f( UDelta;Eg1(x)g2(x)...gn(x)...) = rg1(x) + E INFIN; k =2 (kH -1 11 rgk(x) ugi(x) . i=1 We study local and global properties of function f: structural, extremal, differential, integral, and fractal properties.