Is it true that if $X$ is a zero dimensional locally compact Hausdorff topological space, then if $\beta X\setminus X$ is zero dimensional then there is a continuous function from $X$ onto $[0, 1]$?
I tried to prove it but I'm having trouble. I am reading a survey and I think the author uses this "fact". If it's not true, I will edit this post with more info.
Edit: There was a typo in my question. What I really wanted to ask is the following: Is it true that if $X$ is a zero dimensional locally compact Hausdorff topological space, then if $\beta X\setminus X$ is not zero dimensional then there is a continuous function from $X$ onto $[0, 1]$?
What you ask is not true. Consider X the first uncountable ordinal with the order topology. It is a known fact that every continous real-valued function on X is eventually constant. This has two relevant consequences: the Stone-Cech remainder is a single point, hence zero-dimensional, and the range of every continuous real-valued function is countable.