How can the injectivity of $f$ imply that $b$ is not in the image of the boundary of $Q$?

Question

In the book of Analysis on Manifolds by Munkres, at page 65, it is given that

For $B = f(A)$,

The author mentions the Brouwer theorem of invariance of domain in the next chapter, but I cannot understand how the injectivity of $f$ implies that $b$ is not in the image of the boundary of $Q$.

Answer 1

If $f(\partial Q)$ is not disjoint from $b$ then $f(c)=b$ for some $c\in\partial Q$.

But we also have $f(a)=b$ so the injectivity of $f$ then leads to $a=c\in\partial Q$ contradicting that $a$ is an element of the interior of $Q$.

Answer 2

If $b\in f(\operatorname{Bd}Q)$, then $b=f(x)$ for some $x\in\operatorname{Bd}Q$ and so (since $f$ injective) $a(=f^{-1}(b))\in\operatorname{Bd}Q$. But this is not true.