Suppose $f: M\to N$ is a smooth map between two smooth manifolds, with $M$ compact and connected, and suppose there is a dense subset of $f(M)$ where each fiber is connected, then each fiber of $f$ is connected.

If it helps, you can just consider the case where the set of regular values is dense in $f(M)$ and the fiber of each regular value is connected, and you want to prove every fiber of $f$ is connected.