I'm having a trouble calculating the cardinality of the set of all functions $f:\mathbb{R} \longrightarrow \mathbb{R}$ which have at most $\aleph_0$ discontinuities (let's call the set $M$). A hint is given: Map each function $f$, with a countable set of discontinuities, $d$, to the ordered pair $(f|_d, f|_{R\setminus d})$.
I'm not entirely sure how to go on from here. I know I can create a bijective function $h: M\longrightarrow \bigcup\limits_{d\in D} A_d \times B_d$ where $D$ is the set of all countable subsets of $\mathbb{R}$, $A_d$ is the set of all real functions $f_A:d\longrightarrow \mathbb{R}$, and $B_d$ is the set of all functions $f_B:\mathbb{R} \setminus d \longrightarrow \mathbb{R}$ which do not have discontinuities (I'm not sure if the definition for $B_d$ is correct). Assuming it is correct, I know that $|A_d|=\aleph, \ \ \forall d\in D$, but I do not know how to calculate the cardinality of $B_d$.
I would appreciate any help as to how to go on from here, or hints for better methods.