A note on when monotonicity implies differentiability a.e

When investigating utilities, one of the most frequent assumptions is that of monotonicity, which on the real line implies differentiability almost everywhere (a.e.). In this note we examine this well-known property (due to Lebesgue) of real functions (utility functions). We prove that this result remains valid if a real function is monotonic except for a (countable) set of isolated points. Differentiability a.e. also holds if we augment the set of isolated points with their points of accumulation given that those are isolated from each other. If monotonicity fails on a set of points which is dense on a set with positive measure then the differentiability result is no longer necessarily valid.