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Math Help - Counter Example for Theorem about Monotonic Functions

  1. #1
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    Counter Example for Theorem about Monotonic Functions

    I'm trying (and failing) to find a counterexample to the following false theorem:

    For all functions f,g from Reals to Reals, if g \circ f is monotonic, then at least one of f or g is a monotonic function.
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  2. #2
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    How about f(x) = asin(1/x) and g(x) = 1/x? Or are this f and g monotonic?
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  3. #3
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    I guess this will work, but since asin(1/x) is undefined from x = (-1, 1) it technically isn't a function on the reals. It's only a function from R - (-1, 1) to R.

    If I come up with no other examples this will do, but I feel there should be an example with a true function from R to R
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  4. #4
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    How about f(x) = 1 if x is rational, -1 if x is irrational, and g(x) = x^2?
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  5. #5
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    I was tryng to come up with something at least mostly continuous.
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