Two continuous maps agree on an open dense set, must they be equal?

Question

Suppose $f,g:X\to Y$ are two continuous maps of topological spaces, and $A$ is an open dense subset of $X$, if $f|_A=g|_A$ and $f|_A$ is an open immersion, do we have $f=g$?

Answer 1

No, the property that you need for this to work is Hausdorfness for topological spaces or separatedness for schemes. A counterexample in both categories:

Let $X$ be the affine line with double origin and let $A$ be the complement of one of the origins. Let $f$ be the identity on $X$ and let $g$ be the identity on $A$ and send the second origin to the first one.