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Math Help - Real valued function on [0,1]

  1. #1
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    Real valued function on [0,1]

    I'm stuck on the following question:

    Let A be the set of all real-valued functions on [0,1]. show that there does nor exist a function from [0,1] onto A.

    I don't even know where to start.
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  2. #2
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    Quote Originally Posted by lllll View Post
    Let A be the set of all real-valued functions on [0,1]. show that there does nor exist a function from [0,1] onto A.
    This is a rather advanced theorem.
    So I have no way of knowing what you have already proved.
    But here are the lemmas you will need.

    1) There is an injection f:X \to Y if and only if there is a surjection g:Y \to X.
    2) There is no surjection f:X \to P(X), (power set of X).
    3) If 2^X  = \left\{ {f:X \mapsto \{ 0,1\} } \right\}, the set of functions from X having only two values: 0 or 1.
    Then 2^X  \sim P(X)
    Now clearly the set 2^{[0,1]}  \subseteq A = \left\{ {f:[0,1] \to \mathbb{R}} \right\}

    If there were \varphi :\left[ {0,1} \right]\xrightarrow{{surj}}A then
    \varphi :\left[ {0,1} \right]\xrightarrow{{surj}}2^{[0,1]}  \sim P\left( {\left[ {0,1} \right]} \right).
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