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Math Help - an interesting function

  1. #1
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    an interesting function

    We are given the function f:R -> R defined as follows,

    f(x) = 1/n, if x is a rational number
    f(x) = 0, if x is irrational

    (a) first I need to find a sequence f_n of cont. functions such that f_n(x) -> f(x) as x -> infinity

    (b) second I need to prove that f is continuous at each irrational point and discontinuous at each rational point.

    help?
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  2. #2
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    Quote Originally Posted by matzerath View Post
    We are given the function f:R -> R defined as follows,

    f(x) = 1/n, if x is a rational number
    Is n just some fixed constant here, or is it supposed to be some enumeration of rationals?
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  3. #3
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    Do you mean that f(m/n) = 1/n where m/n is a rational in lowest terms? ie is it Thomae's function - Wikipedia, the free encyclopedia ?
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  4. #4
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    oh, sorry... so if we express Q as a countable union (i.e. Q=U(r_n))
    f(x) = 1/n whenever x = r_n
    f(x) = 0 whenever x is irrational
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