Ishiharaʼs boundedness principle was introduced in Ishihara (1992) [5] and has turned out to be most useful for constructive analysis, see e.g. Ishihara (2001) [6]. It is equivalent to the statement that every sequentially continuous function from to is continuous w.r.t. the usual metric topology on . We construct models for higher order arithmetic and intuitionistic set theory in which both every function from to is sequentially continuous and in which the axiom of choice from to holds. Since the latter is known to be inconsistent with the statement that all functions from to are continuous these models refute .