Nancy Dykes says in the proof of Theorem 3.4 in her article [Generalizations of realcompact spaces][1] that by a result of
John Mack, if for every $p\in \beta X\setminus X$ there exists a nonnegative
upper semicontinuous function $f$ on $\beta X$ such that $f$ is positive on $X
$ and $f\left( p\right) =0$, then $X$ is realcompact.

I looked at both of John Mack's articles in the references but couldn't find
this result. How can I prove this result?


  [1]: https://msp.org/pjm/1970/33-3/pjm-v33-n3-p05-s.pdf