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We wish to thank Krzysztof Ostaszewski for bringing to our attention several of the questions we have considered here.
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Ciesielski, K., Larson, L. The space of density continuous functions. Acta Math Hung 58, 289–296 (1991). https://doi.org/10.1007/BF01903959
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DOI: https://doi.org/10.1007/BF01903959