Functions with connected graphs and $B_1$-retracts - Karlova, O. (Carpathian Mathematical Publications, 2014)
A subset $E$ of a topological space $X$ is called a $B_1$-retract if there exists a mapping $r:X\to E$ which is the pointwise limit of a sequence of continuous mappings $r_n:X\to E$ and $r(x)=x$ for all $x\in E$. We prove that if $Y$ is a union of an increasing sequence of continuums, then the graph of a function $f:\mathbb R\to Y$ is a $B_1$-retract of $\mathbb R\times Y$ if and only if $f$ is continuous.