Analysis provides a way to construct nondecreasing absolutely continuous functions with infinite derivatives in sets of measure zero, highlighting their finite Lipschitz behavior elsewhere.
Given an Fσδ subset in the real line of measure zero, we explain how to obtain a nondecreasing absolutely continuous function such that the derivative is infinite at every point in the set and the little Lipschitz constant (the lower scaled oscillation) is finite at each point outside the set.
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Rmoutil et al. (2025) studied this question.
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