# Decimal representations and continuity

Let I = {0, 1, 2, 3, ..., 9}, and let A be the set of decimal representations of real numbers in the open interval (0,1).

Define the function f: (0, 1) -> A as follows: For x in the open interval (0,1), let f(x) be the element of A with the property that for every positive integer k, the digit i_k in the kth decimal place of f(x) is the least i in I such that x has a decimal representation in which i is the digit in the kth decimal place.

Show that f is discontinuous at every real number x in the open interval (0,1) such that x can be represented as a terminating decimal.

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#### Solution Summary

A complete, detailed proof is provided in the attached .pdf file.

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