Let h be a computable function from the natural number to the rational numbers. A real number x is called h-monotonically computable (h-mc, for short) if there is a computable sequence (x(s)) of rational numbers which converges to x h-monotonically in the sense that the distance of x from x(m) over the distance of x from x(n) is less than h(n) for all m>n. In this paper we investigate classes of h-mc real numbers for different computable functions h.