The main result of the paper under review is that for every Borel function \(f\) from an analytic space \(X\) into a separable metrizable \(Y\), either \(f\) is countably continuous (i.e., \(f\) can be written as a countable union of partial continuous functions), or else it contains a copy of the Pawlikowski function \(P\), a somewhat canonical example of a Baire class \(1\) function which is not countably continuous. This extends a dichotomy discovered by Solecki for Baire class \(1\) functions \(f\) (see [S. Solecki, J. Am. Math. Soc. 11, No. 3, 521–550 (1998; Zbl 0899.03034)]) and later generalized by Zapletal to Borel functions \(f\) defined on Borel subsets of the Baire space [J. Zapletal, Mem. Am. Math. Soc. 793, 140 p. (2004; Zbl 1037.03042)]. The proof notably uses only elementary topology and combinatorics, thus avoiding the use of the sophisticated methods of mathematical logic employed in both [Zbl 0899.03034] and [Zbl 1037.03042]. As an application of their main result, the authors provide partial extensions to all finite levels of a difficult theorem of Jayne and Rogers asserting that a function \(f : X \to Y\) is such that the \(f\)-preimage of every \(\pmb{\Pi}^0_2(Y)\) set is in \(\pmb{\Pi}^0_2(X)\) if and only if \(f\) can be written as a countable union of continuous (partial) functions with closed domains.