Realizability models refuting Ishihara's boundedness principle

Ishihara’s Boundedness Principle BD-N was introduced in [Ish92] and has turned out to be most useful for constructive analysis, see e.g. [Ish01]. It is equivalent to the statement that every sequentially continuous function from NN to N is continuous w.r.t. the usual metric topology on NN. We construct models for higher order arithmetic and intuitionistic set theory in which both every function from NN to N is sequentially continuous and in which the axiom of choice from NN to N holds. Since the latter is known to be inconsistent with the statement that all functions from NN to N are continuous these models refute BD-N.