I've got a question about a lemma in Milnors "TOPOLOGY FROM THE DIFFERENTIABLE VIEWPOINT".
LEMMA 4: If $y\in N$ is a regular value, both for $f$ and for the restriction $f|_{\partial X}$, then $f^{-1}(y)\subset X$ is a smooth $m-n$-manifold with boundary. Furthermore the boundary $\partial (f^{-1}(y))$ is precisely equal to the intersection of $f^{-1}(y)$ with $\partial X$
Maybe this lemma can be called Submersion Theorem for manifold with boundary. So, I want to find a example and take $X=D^2$, $N=\mathbb{R}$ and $f^{-1}(y)$ should be $D^1$. $D^n$ is the unit disk. But I am not able to find the explicit function $f$ and the $y$.
Can someone help me?