Let $X$,$Y$ be compact, connected, smooth manifolds of the same dimension. Suppose you have a surjective smooth map $f : X \rightarrow Y$, such that $|f^{-1}(p) | \leq k$ for all $p \in Y$.

Let $U \subset X$ be an open dense subset.

Question: Is it possible to find $p \in Y$ such that $f^{-1}(p) \subset U$?