On a semicontinuous function (Reprint) (Q2510756)

The authors prove that there exists a semi-continuous function \(f: [0,1] \to \mathbb{R}\) such that \(f\) is essentially discontinuous on the interval \([0,1],\) i.e., the interval \([0,1]\) cannot be partitioned into countably many sets on each of which \(f\) is continuous. In this way N. N. Luzin's question: Does there exist an essentially discontinuous Borel function? is resolved positively. Furthermore, the obtained result is a strengthening of the work given in [\textit{L. Keldysh}, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 1934, No. 4, 192--197 (1934; JFM 60.0972.04)]. The original Russian text was first published in Uch. Zap. Mosk. Gos. Ped. Inst. im. V.I. Lenina 138, No. 3, 3--10 (1958).