I am reading "transversal pre-image of a manifold with boundary" from Differential Topology by M. Hirsch. I have some confusion regarding Theorem 4.2. on the page 31.
Notice that the first highlight says ''theorem for $\partial$-manifold'' but Theorem 4.2. doesn't mention whether $M$ is $\partial$-manifold or not (i.e., $\partial M$ is non-empty or empty). Though in theorem 4.1., it is explicitly mentioned $M$ is a $\partial$-manifold.
Question 1: Does it mean that I can take $\partial M$ is non-empty as well as empty both in theorem 4.2.?
Let $M:=\big\{(x,y)\in \Bbb R^2:y\geq 0\big\}$, $N:=\partial M=\Bbb R\times 0$, $A=[0,1]\times 0$, and $f\colon M\to N$ be the projection map $M\ni (x,y)\longmapsto x\in N$. Certainly, $A\subset N\backslash \partial N$ and $\dim T_a A=1=\dim T_a N$ for each $a\in A$. That is $f\pitchfork A$ and $f\big|\partial M\pitchfork A$. But, $\partial f^{-1}(A)=(0\times \Bbb R)\cup A\cup (1\times \Bbb R)\neq (0\times \Bbb R)\cup (1\times \Bbb R)=f^{-1}(\partial A)$
Question 2: Shouldn't it be $\partial f^{-1}(A)=\big(f^{-1}(A)\cap \partial M\big)\cup f^{-1}(\partial A)$? Of course, if $\partial M=\varnothing$, then it is correct.
The following picture is taken from Differential Topology by Alan Pollack and Victor Guillemin which contains the special case, namely when $\partial A=\varnothing$. See page 60. 