A is a non-empty set. Is there a bijective function $\displaystyle \displaystyle f:A\mapsto A$ such that there exists $\displaystyle H\subset A, H\neq\varnothing $, with $\displaystyle \displaystyle f(H)\subset$ H, and $\displaystyle g:H\mapsto H$, $\displaystyle g(x)=f(x)$, $\displaystyle x\in H$ is not bijective?

Well, uhm, honestly, I have no idea what to do -_- Should I try to "build" a function?

Thanks in advance.