When are Borel functions Baire functions?

The following two theorems give the flavour of what will be proved. Theorem. Let Y be a complete metric space. Then the families of first Baire class functions and of first Borel class functions from [0,1] to Y coincide if and only if Y is connected and locally connected.} Theorem. Let Y be a separable metric space. Then the families of second Baire class functions and of second Borel class functions from [0,1] to Y coincide if and only if for all finite sequences ${\displaystyle U_1,...,U_q}$ of nonempty open subsets of Y there exists a continuous function ϕ:[0,1] → Y such that ${\displaystyle {\varphi }^{-1}\left(U_i\right)\ne Ø}$ for all i ≤ q.