Murugan, V.Palanivel, R.2026-02-052021Real Analysis Exchange, 2021, 46, 1, pp. 51-811471937https://doi.org/10.14321/realanalexch.46.1.0051https://idr.nitk.ac.in/handle/123456789/23406There are continuous functions with complicated yet interesting sets of non-isolated non-strictly monotone points. This paper aims to characterize the sets of isolated and non-isolated non-strictly monotone points of the composition of continuous functions. Consequently, an uncountable dense set of measure zero in the real line and whose complement is also uncountable and dense is obtained. © 2021 Michigan State University Press. All rights reserved.Cantor setIterative rootNon-isolated non-strictly monotone pointsNon-strictly monotone pointsUncountable Measure zero dense set