Abstract
We consider approximately ϕ-homogeneous mappings almost everywhere, that is functions F such that the difference F(αx) − ϕ(α)F(x) is in some sense bounded almost everywhere in a product space. We will prove, under some assumptions, that then either we have some kind of boundedness of ϕ and F, or there exist a homomorphism \( \tilde \phi \) and a \( \tilde \phi \)-homogeneous function \( \overline F \), which are almost everywhere equal to ϕ and F, respectively. From this result we derive the superstability effect for the multiplicativity almost everywhere.
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Jabłoński, W. Stability of homogeneity almost everywhere. Acta Math Hung 117, 219–229 (2007). https://doi.org/10.1007/s10474-007-6092-8
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DOI: https://doi.org/10.1007/s10474-007-6092-8