{'fraction_non_alphanumeric': 0.017054263565891473, 'fraction_numerical': 0.0031007751937984496, 'mean_word_length': 4.295081967213115, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On Certain Dense, Uncountable Subsets of the Real Line

Abstract It is a challenge to define an uncountable set of real numbers that is dense in the real line and whose complement is also uncountable and dense. In this article we specify, for any positive integer k, a partition of the real line into k + 1 sets that are uncountable and dense; moreover, every real number is a condensation point for each of these sets. We show that one of these sets has measure one when restricted to the unit interval, while each of the others has measure zero. We then define a function that is continuous on the set of measure 1 but discontinuous elsewhere.