PERFECTLY EVERYWHERE SURJECTIVE BUT NOT JONES FUNCTIONS

Given a function f : R → R we say that • f is perfectly surjective (f ∈ PES) if f[P] = R for every perfect set P; • f is a Jones function (f ∈ J) if C ∩ f is non-empty for every closed C ⊂ R^2 with dom(C) of size c. M. Fenoy-Munoz, J.L. Gamez-Merino, G.A. Munoz-Fernandez and E. Saez-Maestro in the paper "A hierarchy in the family of real surjective functions" [Open Math. 15 (2017), 486–501] asked about the lineability of the set PES \ J. Answering this question we show that the class PES \ J is c^+ -lineable. Moreover, if 2^

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