Let $X$ be a set whose cardinality is at most that of the continuum. Let $(f^t)_{t > 0}$ be a $(0,\infty)$-indexed family of functions $f^t\colon X \to X$ with the property that $f^{s+t}=f^t \circ f^s$ for all $s,t > 0$.

Suppose we have $x \in X$ and a strictly decreasing sequence $t_n \to 0$ such that $f^{t_n}(x)=x$ for all $n$. Does it necessarily follow that $f^t(x)=x$ for all $t>0$? If it does not follow in general, does it necessarily follow in the case that $X$ is a finite set?