Suppose f: M--->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.

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.

Suppose f: M--->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.

Suppose f: M--->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.