Explore BrainMass
Share

Explore BrainMass

    Decimal representations and continuity

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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.

    © BrainMass Inc. brainmass.com October 10, 2019, 5:02 am ad1c9bdddf
    https://brainmass.com/math/fractions-and-percentages/decimal-representations-continuity-489534

    Solution Summary

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

    $2.19