A simple question. Let $ f:X\to Y $ be a function and let $ E_f:=\{(x, y): f (x)=f (y)\}\subset X\times X $. What is the name of the set $ E(f) $? It would be nice to have some reference also. It seems to be a well known notion/construction in set theory or topology, but I couldn't find anything about it.

Another question is whether the following statement is true: If $ X$ is a connected topological space and the function $ f$ is continuous then $ E_f $ is connected.