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$\sup_{-\infty means?

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 $f(t)$?

1 Answers 1

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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 $\sup \{f(t) | t \in (-\infty,x) \} \neq f(\sup \{t | t \in (-\infty,x) \})$.

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    p.s. In the book it assumed$f$is increasing x_1 < x_2 \implies f(x_1) \leq f(x_2). So you are right. Thanks.2012-09-23