How can I show rigorously the following which is intuitively clear: Given a smooth function $f: \mathbb R \to \mathbb R$ that is finite-to-one(or which satisfy some other from of 'weak injectivity') and a set $\Omega \subset \mathbb R$ of Lebesgue measure zero, then $f^{-1}(\Omega)$ also has measure zero.
Edit: For continuous functions, as i asked before, it is not true.