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Over-iterates of Bernstein's Operators: A Short and Elementary Proof

5. CONCLUDING REMARKS. The motivating question of this note, as to whether there is a modification of Thomae's function that is differentiable on the irrationals, arose while the first author was teaching a one-semester course in real analysis. This question's connection to Diophantine approximation theory was not discovered until after the realization of Proposition 3.2. After the submission of the current manuscript, the authors were informed that a slightly less general version of Proposition 4.2 can be found in [9, p. 232].