Let $X$ be a compact $T_1$ topological space consisting of more than one point, and suppose that $X$ is locally compact (i.e. every point has a local base of compact neighbourhoods), second countable, and is a Baire space. Does $X$ admit a non-constant continuous function into the real numbers?

[In C$^*$-algebra terms, let $A$ be a non-primitive separable C$^*$-algebra whose primitive ideal space is compact and $T_1$. Does the multiplier algebra of $A$ have non-trivial centre?]