Limits of monotone function
$begingroup$
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 $xin(a,b) =I$ 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
$$f(x+):=lim_{ydownarrow x} f(y)$$ 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.
limits supremum-and-infimum monotone-functions
asked Dec 22 '18 at 14:34
Jesper HybelJesper Hybel
1178
$endgroup$
add a comment |
$begingroup$
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 $xin(a,b) =I$ 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
$$f(x+):=lim_{ydownarrow x} f(y)$$ 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.
limits supremum-and-infimum monotone-functions
asked Dec 22 '18 at 14:34
Jesper HybelJesper Hybel
1178
$endgroup$
$begingroup$
Maybe left and right limit?
$endgroup$
– model_checker
Dec 22 '18 at 15:12
add a comment |
$begingroup$
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 $xin(a,b) =I$ 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
$$f(x+):=lim_{ydownarrow x} f(y)$$ 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.
limits supremum-and-infimum monotone-functions
asked Dec 22 '18 at 14:34
Jesper HybelJesper Hybel
1178
$endgroup$
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 $xin(a,b) =I$ 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
$$f(x+):=lim_{ydownarrow x} f(y)$$ 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.
limits supremum-and-infimum monotone-functions
limits supremum-and-infimum monotone-functions
asked Dec 22 '18 at 14:34
Jesper HybelJesper Hybel
1178
asked Dec 22 '18 at 14:34
Jesper HybelJesper Hybel
1178
edited Dec 22 '18 at 16:37
Jesper Hybel
asked Dec 22 '18 at 14:34
Jesper HybelJesper Hybel
1178
asked Dec 22 '18 at 14:34
Jesper HybelJesper Hybel
1178
asked Dec 22 '18 at 14:34
Jesper HybelJesper Hybel
1178
1178
$begingroup$
Maybe left and right limit?
$endgroup$
– model_checker
Dec 22 '18 at 15:12
add a comment |
$begingroup$
Maybe left and right limit?
$endgroup$
– model_checker
Dec 22 '18 at 15:12
$begingroup$
Maybe left and right limit?
$endgroup$
– model_checker
Dec 22 '18 at 15:12
$begingroup$
Maybe left and right limit?
$endgroup$
– model_checker
Dec 22 '18 at 15:12
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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 ${gamma}^+ = overline{limlimits_{x rightarrow +c}};f(x)$ 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 $x.$
answered Dec 22 '18 at 15:23
Dave L. RenfroDave L. Renfro
25.4k34082
$endgroup$
$begingroup$
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.
$endgroup$
– Jesper Hybel
Dec 22 '18 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)
$endgroup$
– Jesper Hybel
Dec 22 '18 at 16:16
1
$begingroup$
@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)
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:34
1
$begingroup$
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 $x.$
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:38
1
$begingroup$
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 $x=c.$
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:49
|
show 1 more comment
$begingroup$
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 $f(x+)=lim_{ydownarrow x} f(y)$.
It is similar for the left limit;
$f(x-)=lim_{yuparrow x} f(y)$
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.
answered Dec 22 '18 at 15:30
CoupeauCoupeau
1296
$endgroup$
$begingroup$
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.
$endgroup$
– Jesper Hybel
Dec 22 '18 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 '18 at 18:36
add a comment |
Your Answer
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3049508%2flimits-of-monotone-function%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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 ${gamma}^+ = overline{limlimits_{x rightarrow +c}};f(x)$ 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 $x.$
answered Dec 22 '18 at 15:23
Dave L. RenfroDave L. Renfro
25.4k34082
$endgroup$
$begingroup$
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.
$endgroup$
– Jesper Hybel
Dec 22 '18 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)
$endgroup$
– Jesper Hybel
Dec 22 '18 at 16:16
1
$begingroup$
@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)
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:34
1
$begingroup$
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 $x.$
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:38
1
$begingroup$
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 $x=c.$
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:49
|
show 1 more comment
$begingroup$
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 ${gamma}^+ = overline{limlimits_{x rightarrow +c}};f(x)$ 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 $x.$
answered Dec 22 '18 at 15:23
Dave L. RenfroDave L. Renfro
25.4k34082
$endgroup$
$begingroup$
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.
$endgroup$
– Jesper Hybel
Dec 22 '18 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)
$endgroup$
– Jesper Hybel
Dec 22 '18 at 16:16
1
$begingroup$
@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)
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:34
1
$begingroup$
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 $x.$
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:38
1
$begingroup$
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 $x=c.$
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:49
|
show 1 more comment
$begingroup$
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 ${gamma}^+ = overline{limlimits_{x rightarrow +c}};f(x)$ 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 $x.$
answered Dec 22 '18 at 15:23
Dave L. RenfroDave L. Renfro
25.4k34082
$endgroup$
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 ${gamma}^+ = overline{limlimits_{x rightarrow +c}};f(x)$ 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 $x.$
answered Dec 22 '18 at 15:23
Dave L. RenfroDave L. Renfro
25.4k34082
answered Dec 22 '18 at 15:23
Dave L. RenfroDave L. Renfro
25.4k34082
answered Dec 22 '18 at 15:23
Dave L. RenfroDave L. Renfro
25.4k34082
answered Dec 22 '18 at 15:23
Dave L. RenfroDave L. Renfro
25.4k34082
25.4k34082
$begingroup$
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.
$endgroup$
– Jesper Hybel
Dec 22 '18 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)
$endgroup$
– Jesper Hybel
Dec 22 '18 at 16:16
1
$begingroup$
@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)
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:34
1
$begingroup$
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 $x.$
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:38
1
$begingroup$
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 $x=c.$
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:49
|
show 1 more comment
$begingroup$
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.
$endgroup$
– Jesper Hybel
Dec 22 '18 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)
$endgroup$
– Jesper Hybel
Dec 22 '18 at 16:16
1
$begingroup$
@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)
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:34
1
$begingroup$
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 $x.$
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:38
1
$begingroup$
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 $x=c.$
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:49
$begingroup$
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.
$endgroup$
– Jesper Hybel
Dec 22 '18 at 16:08
$begingroup$
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.
$endgroup$
– Jesper Hybel
Dec 22 '18 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)
$endgroup$
– Jesper Hybel
Dec 22 '18 at 16:16
$begingroup$
Nevermind I think the identities are simply the definitions, which I guess is consistent with your answer? (I need to learn how to read)
$endgroup$
– Jesper Hybel
Dec 22 '18 at 16:16
1
1
$begingroup$
@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)
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:34
$begingroup$
@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)
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:34
1
1
$begingroup$
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 $x.$
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:38
$begingroup$
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 $x.$
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:38
1
1
$begingroup$
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 $x=c.$
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:49
$begingroup$
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 $x=c.$
$endgroup$
– Dave L. Renfro
Dec 22 '18 at 17:49
|
show 1 more comment
$begingroup$
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 $f(x+)=lim_{ydownarrow x} f(y)$.
It is similar for the left limit;
$f(x-)=lim_{yuparrow x} f(y)$
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.
answered Dec 22 '18 at 15:30
CoupeauCoupeau
1296
$endgroup$
$begingroup$
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.
$endgroup$
– Jesper Hybel
Dec 22 '18 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 '18 at 18:36
add a comment |
$begingroup$
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 $f(x+)=lim_{ydownarrow x} f(y)$.
It is similar for the left limit;
$f(x-)=lim_{yuparrow x} f(y)$
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.
answered Dec 22 '18 at 15:30
CoupeauCoupeau
1296
$endgroup$
$begingroup$
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.
$endgroup$
– Jesper Hybel
Dec 22 '18 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 '18 at 18:36
add a comment |
$begingroup$
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 $f(x+)=lim_{ydownarrow x} f(y)$.
It is similar for the left limit;
$f(x-)=lim_{yuparrow x} f(y)$
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.
answered Dec 22 '18 at 15:30
CoupeauCoupeau
1296
$endgroup$
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 $f(x+)=lim_{ydownarrow x} f(y)$.
It is similar for the left limit;
$f(x-)=lim_{yuparrow x} f(y)$
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.
answered Dec 22 '18 at 15:30
CoupeauCoupeau
1296
answered Dec 22 '18 at 15:30
CoupeauCoupeau
1296
answered Dec 22 '18 at 15:30
CoupeauCoupeau
1296
answered Dec 22 '18 at 15:30
CoupeauCoupeau
1296
1296
$begingroup$
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.
$endgroup$
– Jesper Hybel
Dec 22 '18 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 '18 at 18:36
add a comment |
$begingroup$
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.
$endgroup$
– Jesper Hybel
Dec 22 '18 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 '18 at 18:36
$begingroup$
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.
$endgroup$
– Jesper Hybel
Dec 22 '18 at 16:12
$begingroup$
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.
$endgroup$
– Jesper Hybel
Dec 22 '18 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 '18 at 18:36
$begingroup$
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 '18 at 18:36
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3049508%2flimits-of-monotone-function%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
Maybe left and right limit?
$endgroup$
– model_checker
Dec 22 '18 at 15:12