Is there an example of a discontinuous function, $F$, defined on some complete subset $X\subset R^n$ such that under some metric $d$, $\sum\limits_{n=1}^\infty \|F^n(x)-F^n(y)\|<\infty$ and
Either for multiple $x_i\in X$ where $i\in I$ and $I$ is any arbitrary indexing set, we have $F(x_i)=x_i$.
Or For all $x\in X, F(x)\neq x$?