Let $R$ be a real closed field and $f: R \to R$ a map. Then let $\textrm{F}(f)$ be the set of semialgebraic subsets of $R^2$, which contain $(t,f(t))$ for all $0< t< \epsilon$ for some $\epsilon >0$. Clearly $\textrm{F}(f)$ is a filter on the set of semialgebraic subsets subsets of $R^2$ and if $f$ is defineable, then $\textrm{F}(f)$ is an ultrafilter. An other example would be the exponential function. Does anyone know for which functions $f$ the set $\textrm{F}(f)$ is also an ultrafilter?