Limits of monotone function

Question

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.

Answer 1

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 $x.$

Answer 2

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.