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What does the notation $f(x^{+})$ and $f(x_+)$ mean? The context is the following

I have a proposition concerning monotonic increasing functions, so $f$ is nondecreasing, also $x\in(a,b) =I$ where $f$ is defined everywhere on $I$ and it is claimed

$$f(x^+)=\lim_{\delta \downarrow 0}\{ \inf_{x<y<x+\delta} f(y)\}$$ and $$f(x_+)=\lim_{\delta \downarrow 0}\{ \sup_{x<y<x+\delta} f(y)\}$$

the author also uses the notation

$$f(x+):=\lim_{y\downarrow x} f(y)$$ but unfortunately does not bother to define $f(x^{+})$ and $f(x_+)$ but he says they are limits.

A reference is given Phillips (1984), Sections 9.1 (p. 243) and 9.3 (p. 253). PHILLIPS, E. R. (1984): An Introduction to Analysis and Integration Theory. Dover Publications, New York.

Unfortunately I do not have access to that book at the moment.

Based on the answer by Dave (see below) I realized that the identities were simply definitions.

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  • $\begingroup$ Maybe left and right limit? $\endgroup$
    – Student
    Dec 22, 2018 at 15:12

2 Answers 2

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Phillips does not use the notation $f(x^{+})$ and $f(x_+),$ at least not on p. 243 or on p. 253. However, on these pages Phillips discusses the four extreme limits of a function $f(x)$ at the point $x=c,$ namely the lower and upper left limits (i.e. left side lim-inf and left side lim-sup) and the lower and upper right limits (i.e. right side lim-inf and right side lim-sup), and she uses the notion ${\gamma}^+ = \overline{\lim\limits_{x \rightarrow +c}}\;f(x)$ for the upper right limit and similarly for the other three extreme limits. (Interestingly, just two weeks ago I happened to praise Phillips' book in a comment.)

Given what you've said, I'm pretty sure that $f(x^{+})$ and $f(x_+)$ are intended to denote, respectively, the upper right and lower right extreme limits of the function $f$ at $x.$

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  • $\begingroup$ Thx. Dave. Sorry I do not have a background in mathematics so I would find it very helpful if you could add a formal definition, then I will accept and upvote. As it is I am having trouble understanding how the "lim-inf" and "lim-sup" you talk about is different from the right hand side identities in my question. If you care to expand I would appreciate it. $\endgroup$ Dec 22, 2018 at 16:08
  • $\begingroup$ Nevermind I think the identities are simply the definitions, which I guess is consistent with your answer? (I need to learn how to read) $\endgroup$ Dec 22, 2018 at 16:16
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    $\begingroup$ @Jesper Hybel: See if this Wikipedia article helps. I can probably find more things to look at if you need them, but one thing that might help is that there are at least three different formulations of "lim-inf" and "lim-sup" (I know of 3 ways, and saying "at least" is my way of being careful), so what in some books will be a definition will be an exercise or theorem in another book. In addition to these different formulations, there are also two DIFFERENT notions of "lim-inf" and "lim-sup" --- the more (continued) $\endgroup$ Dec 22, 2018 at 17:34
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    $\begingroup$ common version in which the value of the function at the point doesn't matter (i.e. take sequences approaching the point in which none of the sequence terms is equal to the point; or use deleted open intervals for the epsilon-delta formulations) and a lesser used version in which the value of the function at the point does matter (see my comments here). The version you're dealing with is the more common version, because you have sequences decreasing/increasing to $x$ and open intervals that do not contain the point $x.$ $\endgroup$ Dec 22, 2018 at 17:38
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    $\begingroup$ Finally, the issue with using "left" and "right" is simply a refinement of the usual notion in which the behavior near the point being investigated is limited to one side of the point (left side or right side). There is an equivalent but somewhat abstract way of thinking about the right limit versions at $x=c$ (I'm specifying one of the sides to make the explanation more concrete), where what you do is to replace the original function with a new function whose domain doesn't include anything to the left side of $x=c.$ $\endgroup$ Dec 22, 2018 at 17:49
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I believe this is the way author marks one-sided limits. (In some literature those one-sided limits might be represented differently.)

$f(x^+)$ is so-called right limit. It is the value function $f$ is approaching as its argument is descending from greater values to $x$.

So $f(x^+)=\lim_{\delta \downarrow 0}\{ \inf_{x<y<x+\delta} f(y)\}$ means that $f(x^+)$ is the right limit.
This is the value you are approaching as you check all the smallest values of function $f$ when you use $f$ on arguments slightly bigger than $x$. (which is the point where you are calculating your right limit.) This is what $x < y < x + \delta$ means.
Simply, looking at the graph, this would be the value function $f$ is approaching where you go from right to $x$.
Commonly used notation for right limit is $f(x+)=\lim_{y\downarrow x} f(y)$.

It is similar for the left limit; $f(x-)=\lim_{y\uparrow x} f(y)$ Now you are cheching values of $f$ when the variable is increasing form smaller values (from left on the number scale) to $x$.

Left and right limit are not always the limit. A function can have only left/right limit, but the limit does not exist in that point. But if there are both, left and right limit and are the same, this is also the limit of the function.

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  • $\begingroup$ Thx, that was also my initial thought, however since the author has explicit notatiton for the right limit as $f(x+)$, and also distinguishes between superscript +, $f(x^+)$, and subscript +, $f(x_+)$, I think it is not the right interpretation. $\endgroup$ Dec 22, 2018 at 16:12
  • $\begingroup$ I see. There is obviously more behind it. It is sometimes hard to figure it out if it is out of context as different authors might use different notations for some things. $\endgroup$
    – Coupeau
    Dec 22, 2018 at 18:36

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