Sierpiński's hierarchy and locally Lipschitz functions

Let Z be an uncountable Polish space. It is a classical result that if I ⊆ ℝ is any interval (proper or not), f: I → ℝ and ${\displaystyle \alpha <{\omega }_1}$ then f ○ g ∈ ${\displaystyle B_{\alpha }\left(Z\right)}$ for every ${\displaystyle g\in B_{\alpha }\left(Z\right){\cap }^{Z}I}$ if and only if f is continuous on I, where ${\displaystyle B_{\alpha }\left(Z\right)}$ stands for the αth class in Baire's classification of Borel measurable functions. We shall prove that for the classes ${\displaystyle S_{\alpha }\left(Z\right)(\alpha >0)}$ in Sierpiński's classification of Borel measurable functions the analogous result holds where the condition that f is continuous is replaced by the condition that f is locally Lipschitz on I (thus it holds for the class of differences of semicontinuous functions, which is the class ${\displaystyle S_1\left(Z\right)}$). This theorem solves the problem raised by the work of Lindenbaum ([L] and [L, Corr.]) concerning the class of functions not leading outside ${\displaystyle S_{\alpha }\left(Z\right)}$ by outer superpositions.