One class of continuous functions related to Engel series and having complicated local properties

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 defined by the Engel representation of argument and some convergent real series. We study local and global properties of this function: structural, extremal, differential, integral, and fractal properties.