Best approximations by rational functions with respect to the Hausdorff distance

Inverse theorems on the best approximations of plane sets in a Hausdorff metric by means of rational functions are cited. It is shown, among other things, that if R_n,r (F, [a, b])=o(1/n), then there exists a set P [a, b] of complete measure over which F constitutes a single-valued function.Translated from Matematicheskie Zametki, Vol. 11, No. 5, pp. 491–498, May, 1972.