Can't you define a discontinuous function over some other set besides the reals, such as the set of all computable numbers? For instance, a step function is discontinuous at a single point. You can also define a function that's discontinuous at every point, such as the indicator function of the rationals.
I don't have any experience with constructive set theory, so maybe I'm missing something.
I don't have any experience with constructive set theory, so maybe I'm missing something.