In Rudin's Real and Complex Analysis,chapter 11,I ran into a puzzle.
Here is some definition:
$\Omega_\alpha$ is the smallest convex open set that contains $D(0;\alpha)$ and has the point 1 in its boundary.
Rotated copies of $\Omega_\alpha$,with vertex at $e^{it}$, will be denoted by $e^{it}\Omega_\alpha$.
If $0<\alpha<1$ and $u$ is any complex function with domain $D(0;1)$,its nontangential maximal function $N_\alpha u$ is defined on $\partial D(0;1)$ by $$ \left( N_{\alpha}u \right) \left( e^{it} \right) =sup\left\{ \left| u\left( z \right) \right|:z\in e^{it}\Omega _{\alpha} \right\} $$Similarly,the radial maximal function of $u$ is $$ \left( M_{rad}u \right) \left( e^{it} \right) =sup\left\{ \left| u\left( re^{it} \right) \right|:0\le r<1 \right\} $$
And the author says that "If $u$ is continuous and $\lambda$ is a positive number,then the set where either of these maximal functions is $\leq \lambda$ is a closed subset of $\partial(D(0;1))$.
I don't know how to prove it?
The difficulty is that the Domain keeps changing,and the domain is not compact so I don't know how to use the continuity of $u$.
I do hope someone can teach me how to solve it.