A Characterization of Bimeasurable Functions in Terms of Universally Measurable Sets

The purpose of this note is to show, assuming the continuum hypothesis, that a Borel function, $f$, mapping a Borel subset, ${D_f}$, of a separable complete metric space, ${M_1}$, into a separable complete metric space, ${M_2}$, maps Borel subsets of ${D_f}$ onto Borel subsets of ${M_2}$ if, and only if, $f$ maps universally measurable subsets of ${D_f}$ onto universally measurable subsets of ${M_2}$.