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→ E which is the pointwise limit of a sequence of continuous mappings r_n:X→ E and r(x)=x for all xE. We prove that if Y is a union of an increasing sequence of continuums, then the graph of a function f:R→ Y is a B_1-retract of R× Y if and only if f is continuous.