$\sup_{-\infty
Does it mean the least upper bound of the set of $f(t)$
OR the least upper bound of $t$ which will then be applied to $f(t)$?
$\sup_{-\infty
Does it mean the least upper bound of the set of $f(t)$
OR the least upper bound of $t$ which will then be applied to $f(t)$?
It means the least upper bound of the set $\{f(t) | t \in (-\infty,x) \}$.
To see the difference, consider $f(x) = \arctan (-x)$. $f$ is strictly monotone decreasing. Then you can see that $\sup \{t | t \in (-\infty,x) \} = x$, but $\sup \{f(t) | t \in (-\infty,x) \} = 1$.
This also illustrates that $\sup \{f(t) | t \in (-\infty,x) \} \neq f(\sup \{t | t \in (-\infty,x) \})$.