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Fix $x_0\in \partial \Omega$, $\epsilon >0$, and the power barrier for an exterior cone $g$ from part (i). Write $M=\Vert f{\Vert }_{\infty }$. By continuity of the boundary data, there is $\rho >0$ such that $|\phi (x)-\phi (x_0)|<\epsilon$ on $\partial \Omega \cap B_{\rho }(x_0)$. On the rest of the boundary, $-g\ge m{\rho }^{\nu }$. The boundary is compact, so $\phi$ is bounded. Choose $A>0$ large enough that $A\delta \ge M$ andThen the Perron subfunction for the Poisson equation and its superfunction counterpartsatisfy $\Delta v_{-}\ge M\ge f$, $\Delta v_{+}\le -M\le f$, and $v_{-}\le \phi \le v_{+}$ on the whole boundary. The near-boundary ordering uses the choice of $\rho$; the remaining ordering uses the choice of $A$. In particular the Perron family is nonempty and bounded above.

The Perron method for the Dirichlet problem and the permitted comparison give $v_{-}\le u\le v_{+}$. Since $g(x)\to 0$ as $x\to x_0$, we obtainLetting $\epsilon \downarrow 0$ provesThe interior Perron theorem gives continuity inside $\Omega$, and the displayed limit together with continuity of $\phi$ proves the continuous extension $u\in C^0(\overline{\Omega })$ with the required boundary trace. Thus every boundary point is a regular boundary point.