$(X,d)$ is a compact metric space and $$f\ :\ X\rightarrow X $$ be such that $$d(x,y)=d(f(x),f(y))$$
Show that $f$ is onto .
I have to show that for any $y\in X$ there exist $x\in X$ such that $$f(x)=y.$$
How to begin$?$
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1See http://math.stackexchange.com/questions/34337/isometry-on-compact-space – Lukas Geyer Sep 27 '15 at 15:47
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2http://math.stackexchange.com/questions/170989/a-isometric-map-in-metric-space-is-surjective – Damian Reding Sep 27 '15 at 15:49
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@doglah : How does that answer my question $?$ Where is the contradiction $?$ – user118494 Sep 27 '15 at 16:25
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Sorry, my old comment had the wrong link. I hope this one is more helpful: http://math.stackexchange.com/questions/189490/isometry-fx-to-x-is-onto-if-x-is-compact?lq=1 – Ailbe Sep 27 '15 at 16:26