Murugan, V.Palanivel, R.2026-02-052021Aequationes Mathematicae, 2021, 95, 1, pp. 107-12419054https://doi.org/10.1007/s00010-020-00739-whttps://idr.nitk.ac.in/handle/123456789/23383In this paper, we prove that continuous non-PM functions with non-monotonicity height equal to 1 need not be strictly monotone on its range, unlike PM functions. An existence theorem is obtained for the iterative roots of such functions. We also discuss the Hyers–Ulam stability for the functional equation of the iterative root problem. © 2020, Springer Nature Switzerland AG.Characteristic intervalHyers–Ulam stabilityIterative rootNon-isolated fortNon-monotonicity height