general topology – Degree of a function maps \$Omega subset mathbb{R}^n to mathbb{R}^m\$ for \$n

I am doing self-reading on some degree theory using “Topological Degree Theory and Applications” by Donal O’Regan, Yeol Je Cho, and Yu-Qing Chen; I am totally new to this field and not quite familiar with the notion. We first define the degree of a $$C^1$$ map as follows:

where the $$J_f := det Df$$. I am stuck on this basic theorem:

I do not see how a direct computation works in this case. Could anyone help me with this? Thanks!

Is this bounded convex set in \$mathbb{R}^n\$ closed?

Suppose we have a bounded set $$C subset mathbb{R}^n$$ that is convex and non-empty.
And suppose the family of linear functions $$(f_{x})_{x in mathbb{R}^n}$$ given by $$f_{x}: C rightarrow mathbb{R}$$, $$f_{x}(c) = x.c$$ for $$c in C$$ attain their maximum and minimum in the set $$C$$.

Does this mean $$C$$ is closed (and hence compact) in $$mathbb{R}^n$$?

My idea: I think this does imply $$C$$ is closed but I am not sure how to write my argument “properly”. For every vector $$x in mathbb{R}^n$$, there is a point in $$C$$ that is “furthest” in the direction orthogonal to $$x$$. Then because we are in a convex set we can just “join up all these points” and our set is closed. But how do I write this formally.

Remark: Also is it true that if a linear function on a convex set attains its maximum/minimum it does so on the boundary?