Descriptive mapping properties of typical continuous functions (Q1890639)

In this paper is answered in the affirmative the following question posed by G. Petruska: Is it true that for typical real valued continuous functions \(f\) on \([0, 1]\) there always exists a \(G_ \delta\)-set whose \(f\)-image is not a Borel set? The proof is based on the standard idea to represent analytic sets as projections of \(G_ \delta\) plane sets and to use squarefilling Peano curves. The Banach-Mazur game [\textit{J. C. Oxtoby}, Ann. Math. Stud. 39, 159-163 (1957; Zbl 0078.329)] plays an essential role in the proof of the key proposition.