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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 $x\in X, F(x)\neq x$?

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    @joriki: Sorry for the ambiguities.$I$has to bhave at least 2 elements for the "either" option. Yes,$I$should probably use of instead. Edited. :)2011-12-11

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Yes. Take $X=[0,1]$ and

$F(x)=\begin{cases}\frac12&x=0\;,\\\frac x2&x\ne0\;.\end{cases}$

$F$ is discontinuous at $0$. The sum is a geometric series and thus convergent. And $\forall_{ x\in X}F(x)\ne x$.

Note that the discontinuity at $0$ isn't required to make this work; we could introduce arbitrary discontinuities within the interval, as long as the iteration eventually moves beyond them towards $0$.

Clearly we can't have the other case, $F(x_i)=x_i$ for multiple $x_i\in X$, since in that case the sum would diverge for $x=x_1$, $y=x_2$, being the sum over a non-zero constant.