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It's common to say that constructive mathematics is continuous, as it cannot prove the existence of a discontinuous function. One can certainly reason constructively about the various things traditionally considered continuous mathematics. I find your opposite perspective on it a bit odd.

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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 wouldn't say that it cannot prove the existence of a discontinuous function. Rather it can prove that a point of discontinuity makes it not a function.

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