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Math Help - Inverse image of a function defined on R

  1. #1
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    Exclamation Inverse image of a function defined on R

    The question's listed below:

    Let f be a real-valued function defined on R (real). Show that the inverse image with respect to f of an open set is open, of a closed set is closed, and of a Borel set is Borel.

    Thank you so much for the help!
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  2. #2
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    Re: Inverse image of a function defined on R

    That is not true unless your function is continuous. For example:
    Let f:\mathbb{R} \to \mathbb{R} define the function:
    f(x)=\left\{\begin{array}{lr}x & \text{if x is irrational}\\x+1 & \text{if x is rational}\end{array}\right.

    f^{-1}(1,2) = ((0,1) \cap \mathbb{Q})\cup((1,2) \setminus \mathbb{Q}). But this is the preimage of the open interval (1,2) whose preimage is neither open nor closed.
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