Number of connected components of $f^{-1} (U)$

Question

Let $f:\mathbb{R}^n \to \mathbb{R}$ be an analytical function (semialgebraic,polynomial if needed), $U$ be an open connected subset of $\mathbb{R}$. What can we say about the nuber of connected components of $f^{-1}(U)$? I'm interested in particular if its a finite number or not. I'm writing a prove and it will be helpful if it is a finite number. In my case $\nabla f(x)\neq 0$ if its helpful and i'm only interested in number of connected componnents at infinity (that mean i only care what is happening outside a ball B(0,R) where R>>0) .