Regular boundary point

Remarks

Example: if $\Omega$ is the open unit ball, every boundary point is regular, because $H_f$ is the Poisson integral of $f$, which tends to $f(\xi)$ at each $\xi$. Non-example: for the punctured ball $\Omega = \set{0 < \abs{x} < 1}$, the point $0$ is irregular: for $f = 0$ on the sphere and $f(0) = 1$, one has $H_f = 0$, which does not tend to $1$. Equivalently, $\xi$ is regular if and only if there is a barrier at $\xi$. The Dirichlet problem on $\Omega$ has a solution continuous on $\overline{\Omega}$ for every continuous $f$ if and only if every point of $\partial\Omega$ is regular; the solution is then $H_f$.