What is the definition of $f \in C^{\infty}(\overline{U})$?

Question

I am reading Evans' Partial Differential Equations, Chapter $5$. I am concerned about the notation $f \in C^{\infty}(\overline{U})$, for an open set $U$. Does it mean $f$ is $C^{\infty}(U)$ and that $f $ is continuous in $\overline{U}$?

Answer 1

For various reasons, derivatives are badly behaved on the boundary of regions. Accordingly we usually only define them on open sets, especially in dimension greater than $1$. Yet in PDEs we have to interact with boundaries, so we should assume something about behavior at the boundary as well. The usual way out is to assume that all your derivatives exist and are continuous on the interior and that the function is also continuous on the whole closure (including the boundary). It is more or less the same to assume that your function is actually only defined on $U$ but that it admits a continuous extension to $\overline{U}$. The meaning of this continuity is the same as it is for any mapping between metric spaces.

Answer 2

It means that also the derivatives $f^{(k)}$, defined of $\Omega$, should be uniformly continuous on $\Omega$ so that they can be continuously and uniquely extended to $\overline{\Omega}$.