On the rate of approximation of singular functions by step functions

We consider approximations of a monotone function on a closed interval by step functions having a bounded number of values: the dependence on the number of values of the rate of approximation in the norm of the spaces L _p is studied. A criterion for the singularity of the function in terms of the rate of approximation is obtained. For self-similar functions, we obtain sharp estimates of the rate of approximation in terms of the self-similarity parameters. Functions with arbitrarily fast and arbitrarily slow (down to the theoretic limit) rate of approximation are constructed.