What is happening here : $\partial U $ is flat and $x^o \in \partial U $ is the center of the sphere, assume $\partial U $ to be flat near $x^0 $ and $x_n=0$ I am looking for a concrete example to appreciate whats happening when i define function the following way . Example can be in 1d or 2d . Thanks for ur help.
$U$ is a open, bounded subset of $\mathbb R^n$. I think flat here means that the $\partial U$ divides the ball symmetrically . I am not pretty sure.
The text clip is from Partial Differential Equations by Lawrence C. Evans, p. 225, Chapter 5. In the second edition (p. 269, Section 5.4), $C^\infty$ has been replaced by $C^1$.