I've recently been interested in the following type of functions. A total computable function *f*:**N**→**N** is *effectively closed* if there is a computable function *p* such that *f*[**N** \ W_{e}] = **N** \ W_{p(e)}, where W_{e} is the e-th c.e. set.

Have effectively closed functions been studied? If so, what are they normally called?

I would also appreciate pointers to some uses and/or alternative characterizations of effectively closed functions.

**Motivation.** It is well-known that there is a near-perfect analogy between the adjectives *computable* and *continuous*. For example, a total function *f*:**N**→**N** is computable if and only if it is effectively continuous, i.e. there is a computable function *p* such that *f*^{-1}[W_{e}] = W_{p(e)}. [For the backward implication, let *q* be a computable function such that W_{q(n)} = {n} and use the composite *p*∘*q* to enumerate the graph of *f*.] A similar trick shows that a total function is effectively open if and only if it is computable. However, a total computable function is not necessarily effectively closed since that entails that the range of *f* is computable and, indeed, that *f* maps every computable set onto a computable set. Also, the notion is nontrivial since non-constant polynomials and increasing functions are effectively closed.

**Update.** Joel David Hamkins gave the following characterization of effectively closed computable functions: they are the computable functions *f*:**N**→**N** for which there is a computable *b*:**N**→**N** such that *f*^{-1}(n) ⊆ {0,1,...,b(n)} for every n ∈ **N**. Although I accepted Joel's answer, the main question is still open.