Proof of A Viscosity Solution of the Dirichlet Problem is Continuous and Attains the Boundary Data

Conventions. The order $\le$ of the ordered field $\mathbb{R}$ is a total order, so its antisymmetry is an axiom of that definition. Being a viscosity solution of the Dirichlet problem $(F,g)$, the function $u$ is by definition both a viscosity subsolution and a viscosity supersolution of $(F,g)$; this is used in each of the three arguments below.

Proof of claim 1. Let $x\in\partial\Omega$. Clause Viscosity Sub- and Supersolutions and Solutions of the Dirichlet Problem §subsolution gives $u(x)\le g(x)$ and clause Viscosity Sub- and Supersolutions and Solutions of the Dirichlet Problem §supersolution gives $g(x)\le u(x)$, so $u(x)=g(x)$ by antisymmetry.

Proof of claim 2. By clause Viscosity Sub- and Supersolutions and Solutions of the Dirichlet Problem §subsolution the function $u$ is a viscosity subsolution of $F$ up to the boundary of $\Omega$, hence upper semicontinuous on $\overline{\Omega}$; by clause Viscosity Sub- and Supersolutions and Solutions of the Dirichlet Problem §supersolution it is a viscosity supersolution of $F$ up to the boundary of $\Omega$, hence lower semicontinuous on $\overline{\Omega}$. Let $x\in\overline{\Omega}$. Then $u$ is upper semicontinuous at $x$ relative to $\overline{\Omega}$ and lower semicontinuous at $x$ relative to $\overline{\Omega}$, so claim 2 of Semicontinuity Under Negation and Characterization of Continuity gives that $u$ is continuous at $x$ relative to $\overline{\Omega}$. As $x$ was arbitrary, claim 2 follows.

Proof of claim 3. By clause Viscosity Sub- and Supersolutions and Solutions of the Dirichlet Problem §subsolution and the definition of a subsolution up to the boundary, the restriction $u|_{\Omega}$ is a viscosity subsolution of $F$ on $\Omega$; by clause Viscosity Sub- and Supersolutions and Solutions of the Dirichlet Problem §supersolution it is likewise a viscosity supersolution of $F$ on $\Omega$. By the definition of a viscosity solution, $u|_{\Omega}$ is a viscosity solution of $F$ on $\Omega$. $\blacksquare$