# When does a multiplication operator on $L^2$ have closed range?

I'm working on the following problem in Conway's Functional Analysis. Here $$\phi$$ is a bounded measurable function on $$(X, \Omega, \mu)$$. I was able to answer the first part of the problem but I am stuck on the second. My first idea was to look at the spectrum, as injectivity + closed range $$\implies$$ surjectivity. However, I haven't figured out the case when $$\phi$$ is zero on a set of positive measure. One sufficient condition I came up with is for $$X \setminus ker(\phi)$$ to be a closed set. I don't know if this is also a necessary condition and I would really appreciate some help!