Prove that the set $X = \{x \in R^L_+| u(x) \geq \bar u\}$ is closed Prove that the set $X = \{x \in R^L_+| u(x) \geq \bar u\}$ is closed. 

Saw this statement in the textbook but I'm not sure how this is the case when we don't have any restrictions on $u(x)$ such as continuity. I can prove this if it is continuous, but I'm not sure how to do it if  isn't.
The statement does not seem to be true.

Define $u$ as $u(x) = -1$ if $x \leq 0$, $u(x) = 0$ if $x \in (0,1)$ and $u(x) = 1$ if $x \geq 1$. The set of points $x$ for which $u(x) \geq 0$ is $(0,\infty)$, which is not closed.