"Let $\;U\subset \mathbb R^n\;$ be open and bounded and consider a linear function $\;f:U \to \mathbb R\;$."
Our professor used the above example in the class in order to give us a function which is not necessary zero on $\;\partial U\;$ and now I want to understand why this example was chosen.
As I understand it, for $\;n=1\;$ a non-trivial linear function cannot be zero in both boundary points of the interval $\;U\;$. However for $\;n\gt 1\;$ this argument becomes a little more complicated to me.
I would appreciate if somebody could help me visualize the above in more dimensions.
Thanks in advance!
Consider $f(x,y)=x+y$ and let the $U$ be the triangular region in xy plane bounded by $x+y=10$, $x=0,$ and $y=0.$
$f(x,y)$ is a linear function and $f(x,y)$ on the boundary is not zero except at $(x,y)=(0,0).$