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How to define this function

  1. Feb 14, 2012 #1
    1. The problem statement, all variables and given/known data

    If f(x) < g(x) < h(x) for all x E R, and if f and h are continuous functions, must g also be continuous? If so, why? If not, can you come up with a counter example?

    What do you think?
     
  2. jcsd
  3. Feb 15, 2012 #2
    Hi...
    For the given conditions ,the function g need not be continuous.
    Example : f(x) = sin(x) - 3 , h(x) = sin(x) + 3 , g(x) = sin(1/x)
    for this example , f(x) < g(x) < h(x) . f and h are continuous . But g is not continuous at x tends to 0. Instead of g(x) = sin(1/x) , u can also take a signum function which is not continuous at x = 0.
     
  4. Feb 15, 2012 #3

    HallsofIvy

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    Staff Emeritus
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    Blazeatron's example is sufficient but simpler-
    f(x)= 0 for all x, h(x)= 3 for all x, g(x)= 1 if x is rational, 2 if x is irrational. g is discontinuous for all x.
     
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