Remark

The theorem may be false if we do not suppose that $X$ is a Baire space. For example if, for each rational number $p/q$ ($p, q$ being coprime integers, $q > 0$) we put $f(p/q) = q$, we have a lower semi-continuous function $f$ on the rational line $Q$ which is finite at each point (cf. Chapter IV, § 6, no. 2); but there is no non-empty open set in $Q$ on which $f$ is bounded above.