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This paper obtains estimates on the degree of nearly comonotone approximation which extend and improve the estimate obtained by Newman, Passow, and Raymon. In particular, the restriction that / e LipM 1 is removed, and estimates for the degree of nearly comonotone approximation are obtained for all proper piecewise monotone functions. It is also shown that if / exists and is continuous on the interval, then the ordinary polynomials of best approximation form a nearly comonotone sequence.doi:10.2307/2040212 fatcat:t4wj3b2gxvcyfo25kbqzq7td4a