Evans pdes p27 - Strong maximum principle
Let $u\in C^2(U)\cap C(\bar{U})$ be harmonic within $U$.
Suppose $u(x_0)=\text{max}_{\bar{U}}$ for $x_0\in U$.
Then since $u$ is harmonic, it satisfies the mean value property, and hence there is some ball in $U$ where $$M=u(x_0)=-\!\!\!\!\!\!\int_{B(x_0,r)} u dy \leq M$$
Where the equality only holds if $u(y)=M$ for each $y\in B(x_0,r)$. I want to understand if this is the logic that is wanted for the following sentence:
Then the set $\{x\in U: u(x) =M\},$ is both open and relatively closed.
It is open because this is satisfied on the union of balls around each point that is maximal.
It is relatively closed because $u$ is continuous, and we are taking the preimage of $\{M\}$, a closed set in $\Bbb R^1$.
Secondary question. He says that if $U$ is connected, then the above set is equal to $U$, and hence the result follows. That is fine. But then he says that thus the result follows that $$\text{max}_{\bar{U}} u = \text{max}_{\partial U} u$$
Which if $U$ is connected I agree with. But that equality didn't have $U$ connected as a premise. Did Evans leave this part of the proof open?