Semicontinuous functions that are not countably continuous

Luzin’s theorem on the structure of Lebesgue measurable functions acting fromR intoR is one of the most fundamental statements in real analysis and has numerous applications. Let us recall the formulation of this classical theorem. It is convenient for us to give the formulation in terms of partial functions (cf. the Introduction). As usual, we denote by λ the Lebesgue measure on R.