I'm a bit in truble with this writting : $$x_\varepsilon (t)=\varepsilon t+O(\varepsilon ^2).$$
Does here the rest depend on $t$ ? In other word, I'm not sure if $$x_\varepsilon (t)=t\varepsilon +R(\varepsilon )$$ where $$|R(\varepsilon )|\leq C|\varepsilon |^2$$ or $$|R(\varepsilon )|\leq C_t|\varepsilon |^2$$
or $$x_\varepsilon (t)=t\varepsilon +R(\varepsilon ,t)$$ where $$|R(\varepsilon ,t)|\leq C|\varepsilon |^2$$ or $$|R(\varepsilon ,t)|\leq C_t|\varepsilon |^2,$$
Can someone enlighten me a bit on this point ?
Unfortunately, this depends on context. Some authors will write something like $O_t(\epsilon^2)$ to mean that the constant depends on t. Others will write just as you have written and then write something like "Where the implied constant depends on $t$" or "where the implied constant is independent of $t$" depending on which it is in this case. Do you have the original context of the statement, either the theorem or its proof? If so, that should allow one to figure out from context which they mean.