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Absolute Values and Continuous Functions

  1. Jul 24, 2007 #1
    1. The problem statement, all variables and given/known data

    We recently proved that if a function, f, is continuous, it's absolute value |f| is also continuous. I know, intuitively, that the reverse is not true, but I'm unable to come up with an example showing that, |f| is continuous, b f is not. Any examples or suggestions would be appreciated. Thanks!

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Jul 24, 2007 #2


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    What about [itex]f(x) = -1[/itex] if [itex]x < 0[/itex] and 1 if [itex]x \ge 0[/itex]?
    Then [itex]|f(x)| = 1[/itex] for all x - about as continuous as you get them, but f is not. In fact, you could even do something pathetic like make f equal -1 on all rationals and 1 on all irrational numbers :)
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