Classes of Baire functions

Let $\mathcal{A}$ and $\mathcal{P}$ denote the sets of approximately continuous and almost everywhere continuous functions, and ${B_1}(F)$ denote Baire's first class generated by $F$. The classes ${B_1}(\mathcal{A})$, ${B_1}(\mathcal{P})$, ${B_1}(\mathcal{A} \cap \mathcal{P})$, and Grande's class $\mathcal{A}{\mathcal{P}_1}$ are investigated in some detail. Although Grande's question of whether ${B_1}(\mathcal{A} \cap \mathcal{P}) = {B_1}(\mathcal{A}) \cap {B_1}(\mathcal{A}) \cap \mathcal{A}{\mathcal{P}_1}$ is not settled, we do show, among other results, that $\mathcal{A}{\mathcal{P}_1} \subset {B_1}(\mathcal{P})$.