On the dependence of properties of the graph of a function on the degree of various approximations

This paper studies the dependence of the properties of a function on the degree of its approximation by piecewise monotone functions (in particular, by rational functions and polynomials) in the uniform and Hausdorff metrics, i.e., inverse theorems in the theory of approximation. Of interest are properties related to the Hausdorff dimension and measure of the set of points of discontinuity and to the graph of a function.