Best approximation by integer-valued functions

Given an integer function f, the problem is to find its best uniform approximation from a set K of integer-valued bounded functions. Under certain conditions on K, the best extremal (maximal or minimal) approximation is identified. Furthermore, the operator mapping f to its extremal best approximation is shown to be Lipschitzian with some constant C or optimal Lipschitzian having the smallest C among all such operators. The results are applied to approximation problems.