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Such a map does not exist since a continuous map on $X$ must be constant almost everywhere. In other words, if $f:X\rightarrow X$ is a map such that $f^{-1}[A]\in u$ for each $A\in u$, then $\{x\in X|f(x)=x\}\in u$. For a proof, see the book The Theory of Ultrafilters by Comfort and Negrepontis Thm 9.2 or Andreas Blass's dissertation p. 12.