# Math Help - Inverse image of a function defined on R

1. ## 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!

2. ## 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.