The Borel $\sigma$-algebra $\cal B$ on real numbers has many good properties. For instance, all continuous functions are $\cal B/\cal B$-measurable. On the other side, not only $\cal B$ is not complete but, as discovered by Suslin, $\cal B$ is not closed under *images* of continuous functions.

It is tempting then to consider the $\sigma$-algebra, say ${\cal B}^1$, generated by analytic sets, or even, going up along the projective hierachy, to consider the $\sigma$-algebra ${\cal B}^{\infty}$ generated by projective sets.

It would by nice to know if these $\sigma$-algebra share some properties with the Borel one that make $\cal B$ useful.

Question I:Are continuous functions ${\cal B}^1/ {\cal B}^1$-measurable?