Prove sublevel set is closed

Question

$$f(x)=-\sum_{i=1}^{n}log(b_i-a_i^Tx)$$ $dom\ f_0={Ax<b}, where\ A\ is\ a\ m\ by\ n\ matrix\ with\ rows\ a_i^T$

How to prove that the sublevel sets of $f_0$ is closed?

Generally, for a given function, how to show its sublevel sets are closed or not?