In my analysis text, it seems that $\max$ and $\min$ are replaced by $\sup$ and $\inf$ for 1D single variable function, why is this the case?
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1Related: http://math.stackexchange.com/questions/18605/max-and-min-versus-sup-and-inf?rq=1 – Mankind May 28 '15 at 00:00
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A set can have a greatest lower bound and a least upper bound without having a largest or smallest element in it. E.g. Any open interval in $\Bbb{R}$. – Nick D. May 28 '15 at 00:02
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Consider the maximum value for a function for which the range is $[0,b)$. – Henry Shearman May 28 '15 at 00:02
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@nickD. Can we substitute sub for max and inf for min even if these terms are not interchangeable? – Shamisen Expert May 28 '15 at 00:14
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Yes. Every max is a sup (but not vice-versa), and every min is an inf (but not vice-versa). – Nick D. May 28 '15 at 00:16
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@NickD. So is it common in upper level math to ditch max and min all together? This seems to be the approach of my real analysis book – Shamisen Expert May 28 '15 at 00:17
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It depends on the context. If you're in a situation where either can occur, it would be wrong to write max or min. If you're dealing with, say, finite sets, it would be weird (I think, at least) to write sup or inf. I'm sure there are many people who wouldn't find that weird. – Nick D. May 28 '15 at 00:23
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Because $\max$ only exists if the set contains its $\sup$. The set $\{0.9, 0.99, 0.999, 0.9999, 0.99999,...\}$ has sup 1 but no max.
Giuseppe
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Well there isn't always a maximum. Consider $S=\{r\in\Bbb Q:r<2\}$. This is a bounded set with no maximum element. But we can say that $2$ is the smallest number which is an upper bound of $S$.
Tim Raczkowski
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They're different things. Intuitively, supremum and infimum mean least upper bound and greatest lower bound, respectively, which are different concepts from maximum and minimum. Thus, for example, $(-\infty, 0)$, the set of all negative reals, has a sup of $0$, but no maximum value: whatever negative value $x$ you come up with, $x/2$ is greater but still negative.
Brian Tung
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A maximum and a minimum are attained at some value of the variable (by definition), while the supremum and the infimum (least upper bound and greatest lowerbound) are values which can be approached as close as we wish, but are not necessarily attained.
Example: on $\mathbf R$, the function $\arctan$ has a $\sup$ and an $\inf\,$ ($\frac\pi2$ and $-\frac\pi2$), but there is no $x$ such that $\,\arctan x=\frac\pi2$.
Bernard
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I suppose this depends on what sense/context you mean ‘replace’.. A max is a sup, but the latter always exist if the function is bounded, while the former doesn't always exist. – Bernard May 28 '15 at 00:21