Let $X$ be a compact topological space. Let $f : X \longrightarrow \mathbb { R }$ be a function satisfying $f ^ { - 1 } ( [ n , \infty ) )$ is closed for all $n \in \mathbb { N }$. Pick the correct statements from below.\\
(A) $f$ is continuous.\\
(B) $f ( U )$ is open for each open subset $U$ of $X$.\\
(C) $f ( U )$ is closed for each closed subset $U$ of $X$.\\
(D) $f$ is bounded above.