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

Question

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!