Suppose $\phi: X \rightarrow Y$ and $f:X \rightarrow \mathbb{R}, g:Y \rightarrow \mathbb{R}$ where $X, Y$ are metric spaces and $f, g$ are Baire-$1$ functions. Let $x_0$ be the only point of discontinuity of $f$. Then there exists a sequence $(x_n)_{n \geq 1}$ such that $x_n \rightarrow x_0$.
Collect all such sequences and define $D_{x_0}=\{ (y_n): x_n \rightarrow x_0, \phi(x_n)=y_n \}$. Then the set is a sequence space.
Question: Does there exist an injection $\phi$ such that $\phi(x_0)=y_0 \in Y$ is the point of discontinuity of $g$?
