# Regular selections for multiple-valued functions

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## Abstract

Given a multiple-valued function *f*, we deal with the problem of selecting its single-valued branches. This problem can be stated in a rather abstract setting considering a metric space *E* and a finite group *G* of isometries of *E*. Given a function *f*, which takes values in the equivalence classes of *E*/*G*, the problem consists of finding a map *g* with the same domain as *f* and taking values in *E*, such that at every point *t* the equivalence class of *g*(*t*) coincides with *f*(*t*).

If the domain of *f* is an interval, we show the existence of a function *g* with these properties which, moreover, has the same modulus of continuity of *f*. In the particular case where *E* is the product of *Q* copies of ℝ^{ n } and *G* is the group of permutations of *Q* elements, it is possible to introduce a notion of differentiability for multiple-valued functions. In this case, we prove that the function *g* can be constructed in such a way to preserve *C* ^{ k,α} regularity.

Some related problems are also discussed.

## Keywords

Equivalence Class Related Problem Finite Group Abstract Setting Regular Selection## Preview

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## References

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