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Math Help - properties of continuous functions

  1. #1
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    properties of continuous functions

    1) Show that 2^x = 3x for some x element in (0,1)

    2) Suppose f:[a,b]-->R is continuous and that f([a,b]) is a subset of Q (the rationals). Prove that f is constant on [a,b].
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  2. #2
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    1: Consider f(x)=3x-2^x and then apply the intermediate value theorem to the interval [0,1].

    2: Suppose that f is not constant then it takes on at least 2 different values. Apply the intermediate value theorem and the fact that between any 2 real numbers there is an irrational number.
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