Question: Let $X^m$ and $Y^n$ differentiable manifolds. $f:X\rightarrow Y$ a differentiable map. Show that if $\partial X=\emptyset$, $y\in Reg(f)$ and $f^{-1}(y)\neq\emptyset$, then $y\not\in\partial Y.$
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If there is $y\in\partial Y.$ Then if $F:U\subset\mathbb H^n\rightarrow V$, $p\in U$ such that $F(p)=y$, then $p\in \partial \mathbb H^n$.
Let $x\in f^{-1}(y)$, my idea is to proof that $x\in \partial X$, which is a contradiction.
As $f$ is differentiable, let $G:W\subset\mathbb R^m\rightarrow Z$ a parametrization of $X$, $q\in W$ with $G(q)=x$. Then, $F^{-1}\circ f\circ G$ is differentiable at $q$.
And what else can i do?
Hint: Think about what the map $f$ does to points around $x$. Also, recall what a submersion means for the dimensions of your two spaces.
Addendum:
Everything you did was correct. You just need to push the conclusion out of what you already know. And that comes from the submersion theorem. So we can assume that these open sets you mentioned line up in such a way that $F^{-1}\circ f \circ G (x_1,\ldots,x_m)=(x_1,\ldots ,x_n)$ since $m\geq n$. Then all you have to do is notice that the open set $W$ was open in euclidean space, while $U$ was an open nhood of a boundary point in the half-space. Personally I would say you have reached your contradiction. If you want to be more thorough, you can look at the last coordinate of your points in $U$ and notice they are all positive... I will leave that to you.