Let $f: M \to N$ be a smooth maps between smooth manifolds. Then $f$ is a submersion (by definition) if its differential is also surjective. Now suppose $f$ is surjective. Is it possible that the surjective map $f$ fails to be a submersion on a set in $N$ of measure non-zero? If so, what is such a map?

Suppose the manifolds $M$ and $N$ are non-compact. Does this change the previous answer?

Thanks!