My question: Why is the function $f_\epsilon$ continuous (smooth) up to the boundary?
This is theorem 4.3 in the new edition of Evans & Gariepy's "Measure Theory and Fine Properties of functions."
I understand the proof and constructions. However I have difficulty seeing why the function $f_\epsilon$ is continuous (smooth) up to the boundary -- which is the whole point of the theorem as otherwise this theorem would have nothing more than an earlier theorem claiming approximation by $C^\infty(U)$ functions.
The following question was asked about the same proof, in case it helps:
Question regarding Evan's proof of Global Approximation by $C^∞(\overline{U})$ functions
