The result presented in TFA is that all computable functions over the real numbers are continuous. To generalize that to 'all functions are continuous', you not

The result presented in TFA is that all computable functions over the real numbers are continuous. To generalize that to 'all functions are continuous', you not only need to believe that all functions should be computable, which sounds reasonable for a constructivist, but also that all numbers should be real, which is not. Back when I was in academia, I remember a talk by a mathematician who thought that we should all be doing analysis with rational numbers rather than real numbers. With such a perspective, the sign function defined over rational numbers (e.g. presented as a pair numerator, denominator) is both discontinuous at 0 and computable.