A question about a normal lower semicontinuos function

Question

Theorem: A lower semicontinuos function on a topological space $X$ is normal if and only if for each real number $\lambda $, $\left\{ x\in X:f\left( x\right) <\lambda \right\} $ is a union of regular closed sets.

Suppose that $X$ is a topological space and $D$ is a dense subset of $X$. Also $f:X\longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ is a normal lower semicontinuos function such that $f\left( x\right) $ is positive for each $x\in D$. Then is $f(x)$ greater than or equal to zero for each $x\in X$?

Answer 1

No, consider on the real line $f(x) = 0$ if $x\neq 0$ and $-1$ if $x=0$.