A theorem about countable decomposability

A functionf:X→ ℝ iscountably decomposable(into continuous functions) if the topological spaceXcan be partitioned into countably many setsAnwith each restrictionf│Ancontinuous. According to L. V. Keldysh(2), the question whether every Baire function is countably decomposable was first raised by N. N. Luzin, and answered by P. S. Novikov. The answer is negative even for Baire-1 functions, as is shown in (2) (see also (1). In this paper we develop a characterization of the countably decomposable functions on a separable metric spaceX(see Corollary 1). We deduce that whenXis complete they include all functions possessing the propertyPdefined by D. E. Peek in (3):each non-empty σ-perfect set H contains a point at which f│ H is continuous. The example given by Peek shows that not every countably decomposable Baire-1 function has propertyP.