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 Musta\c{t}\u{a}-Takagi-Watanabe. We also prove that, for a two dimensional standard graded domain $(R, {\bf m})$overafieldofcharacteristic$0$,withgradedideal$I$,if$({\bf m}_p, I_p)$isareductionmod$p$of$({\bf m}, I)$then$c^{I_p}({\bf m}p) \neq c^I{\infty}({\bf m})$implies$c^{I_p}({\bf m}_p)$has$p$ in the denominator.