Ask HN: Can there exist a function whose computability is uncomputable?

[nicholas-cc]

We can mathematically prove that certain mathematical functions and constants are computable or incomputable. For example, we know that the Busy Beaver function is uncomputable, but that arithmetic is computable.

Are there any functions for which their computability has been proven uncomputable, and if so, what are some examples and is there any term for such a function?

[billconan]

https://en.wikipedia.org/wiki/Halting_problem