This is a topological question that came up tangentially to some material I was working on. Suppose $X$ and $Y$ are complete metric spaces and $D$ is a dense subset of $X$. Let $f:D\mapsto Y$ be a continuous injection. Extend $f$ to a function $g:X\mapsto Y$ by continuity. Must $g$ be injective? It seems to me that the answer should be yes, but I haven't been able to prove it.



Injective Function on a Dense Set
Map the open unit interval to a circle minus a point, and then extend it to the closed interval.


