Smooth extension criterion for an immersed submanifold

An injectively immersed submanifold $N\subset M$ is embedded if every smooth function on $N$ extends to a smooth function on $M$. Otherwise a sequence from a different local sheet can converge in $M$ to a point $p$ while remaining outside a small embedded neighborhood of $p$ in $N$; a bump function supported on that neighborhood cannot have a continuous ambient extension.