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.
"Some Observations on `A New Proof of a Theorem of Jayne and Rogers'." Real Anal. Exchange 38 (1) 121 - 132, 2012/2013.