Nondiscreteness of $F$-thresholds

We give examples of two dimensional normal ${mathbb Q}$-Gorenstein graded domains, where the set of $F$-thresholds of the maximal ideal is not discrete, thus answering a question by Mustac{t}u{a}-Takagi-Watanabe. We also prove that, for a two dimensional standard graded domain $(R, {bf m})$ over a field of characteristic $0$, with graded ideal $I$, if $({bf m}_p, I_p)$ is a reduction mod $p$ of $({bf m}, I)$ then $c^{I_p}({bf m}_p) eq c^I_{infty}({bf m})$ implies $c^{I_p}({bf m}_p)$ has $p$ in the denominator.