We prove that every diagonally noncomputable function computes a set $A$ which is bi-immune, meaning that neither $A$ nor its complement has an infinite computably enumerable subset.
Jr., Carl G. Jockusch. Andrew E. M. Lewis. "Diagonally non-computable functions and bi-immunity." J. Symbolic Logic 78 (3) 977 - 988, September 2013. https://doi.org/10.2178/jsl.7803150