It is well-known that a compact Hausdorff $X$ space is scattered if and only if admits no continuous maps onto the unit interval $[0,1]$.


Surprisingly, but I cannot find a good reference to this well-known fact (desirably some textbook). 


In the survey paper "[Scattered spaces](https://books.google.com/books?id=JWyoCRkLFAkC&pg=PA350)" in [Encyclopedia of General Topology](https://www.elsevier.com/books/encyclopedia-of-general-topology/hart/978-0-444-50355-8) this fact is not mentioned, unfortunately.



A reference to a well-known characterization of scattered compact spaces
The proof in the direction that there is no continuous surjection from a compact scattered $X$ onto $[0,1]$ may be found here (Theorem 1):


W. Rudin, *Continuous functions on compact spaces without perfect subsets*, Proc. Amer. Math. Soc. 8 (1957), 39-42.


And the full characterisation may be found in (Theorem 8.5.4, p. 148):


Z. Semadeni, *Banach spaces of continuous functions*, vol. 1, PWN - Polish Scientific Publishers, Warsaw 1971.


