We consider limit periodic sequences and functions with values in pseudometric spaces. We construct limit periodic sequences and functions with given values. For any totally bounded countable set, we find a limit periodic sequence which attains each value from this set periodically. A corresponding result concerning such a construction of limit periodic functions is proved as well. For any totally bounded countable set which is dense in itself, we construct a limit periodic bijective map from the integers into this set. As corollaries, we obtain new results about non-almost periodic solutions of almost periodic transformable difference systems.
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