  We adapt a construction taken from `L. Motto Ros and B. Semmes, A new proof
of a theorem of Jayne and Rogers, Real Anal. Exchange 35(1) (2009/2010),
195-204' in order to correct a mistake contained in the first part of the same
paper. As a byproduct of the new construction, the Jayne-Rogers theorem is
extended to functions whose range is a regular topological space, and a theorem
of Solecki which sharpens the Jayne-Rogers theorem for separable metric spaces
is extended to the non-separable context.
