Under the assumption that all "rules" are recursive (ECT) the statement $\operatorname{Cont}(N^N,N)$ that all functions from $N^N$ to $N$ are continuous becomes equivalent to a statement KLS in the language of arithmetic about "effective operations". Our main result is that KLS is underivable in intuitionistic Zermelo-Fraenkel set theory + ECT. Similar results apply for functions from $R$ to $R$ and from $2^N$ to $N$. Such results were known for weaker theories, e.g. HA and HAS. We extend not only the theorem but the method, fp-realizability, to intuitionistic ZF.