Disjoint Borel Functions

For each $a\in \mathbb{R}$, we define a Borel function $f_a : $\mathbb{R}$ \to $\mathbb{R}$$whichencodes$a$inacertainsense\mathrm{.}WeshowthatforeachBorel$g : $\mathbb{R}$ \to $\mathbb{R}$$,$f_a \cap g = \emptyset$implies$a \in \Delta^1_1©$where$c$isanycodefor$g$\mathrm{.}Wegeneralizethistheoremfor$g$inlargerpointclasses$\Gamma$\mathrm{.}Specifically,if$\Gamma = $\mathbf{\Delta }$^1_2$,then$a \in L[c]$\mathrm{.}Alsoforall$n \in \omega$,if$\Gamma = $\mathbf{\Delta }$^1_{3 + n}$,then$a \in $\mathcal{M}$_{1 + n}©$.