A limit with limit zero everywhere must be zero somewhereDoes there exist a function such that $lim_x to a f(x) = L$ for all $a in mathbb R$ but $f(x)$ is never $L$?Limits of Monotone Functionsuniform convergence, pointwise limitWeak star limitAbout a function finitely valued almost everywhereProve that a function has limit everywhere.Is Thomae's function Riemann integrable on every interval?How to take the derivative of this strange alternative to a devil's slippery staircase?Everywhere Super Dense Subset of $mathbbR$Show that $f$ is Riemann integrable and evaluate $int_0^1 f(x)dx$Prove the characteristic function of Cantor set is continuous almost everywhere.
How difficult is it to simply disable/disengage the MCAS on Boeing 737 Max 8 & 9 Aircraft?
Can a druid choose the size of its wild shape beast?
Who is flying the vertibirds?
Employee lack of ownership
Should we release the security issues we found in our product as CVE or we can just update those on weekly release notes?
Is there a data structure that only stores hash codes and not the actual objects?
Awsome yet unlucky path traversal
How to deal with a cynical class?
Happy pi day, everyone!
What's the meaning of “spike” in the context of “adrenaline spike”?
Are all passive ability checks floors for active ability checks?
Sailing the cryptic seas
Why Choose Less Effective Armour Types?
Identifying the interval from A♭ to D♯
Is it true that good novels will automatically sell themselves on Amazon (and so on) and there is no need for one to waste time promoting?
What exactly is this small puffer fish doing and how did it manage to accomplish such a feat?
Does someone need to be connected to my network to sniff HTTP requests?
Opacity of an object in 2.8
If I can solve Sudoku can I solve Travelling Salesman Problem(TSP)? If yes, how?
In a future war, an old lady is trying to raise a boy but one of the weapons has made everyone deaf
A Cautionary Suggestion
Why doesn't the EU now just force the UK to choose between referendum and no-deal?
PTIJ: Who should I vote for? (21st Knesset Edition)
What should tie a collection of short-stories together?
A limit with limit zero everywhere must be zero somewhere
Does there exist a function such that $lim_x to a f(x) = L$ for all $a in mathbb R$ but $f(x)$ is never $L$?Limits of Monotone Functionsuniform convergence, pointwise limitWeak star limitAbout a function finitely valued almost everywhereProve that a function has limit everywhere.Is Thomae's function Riemann integrable on every interval?How to take the derivative of this strange alternative to a devil's slippery staircase?Everywhere Super Dense Subset of $mathbbR$Show that $f$ is Riemann integrable and evaluate $int_0^1 f(x)dx$Prove the characteristic function of Cantor set is continuous almost everywhere.
$begingroup$
I wish to know if the following is true:
Let $f : [alpha, beta]to mathbb R$ be a function so that
$$ lim_xto x_0 f(x) = 0$$
for all $x_0 in [alpha, beta]$. Then $f(x) = 0$ for some $xin [alpha, beta]$.
The Thomae's function $f: [0,1]to mathbb R$
$$f(x) =begincases 1/q & textif x= p/qin mathbb Q, \ 0 & textotherwise.endcases$$
leads me to the above question. The Thomae function has limit zero everywhere, althought it is nonzero in $mathbb Q$. I think I can take any countable dense subset $Dsubset [alpha, beta]$ and construct a function which is nonzero in $D$ but limit equals zero everywhere. But I can't think of a function that is nonzero everywhere but has zero limit everywhere.
real-analysis limits
$endgroup$
add a comment |
$begingroup$
I wish to know if the following is true:
Let $f : [alpha, beta]to mathbb R$ be a function so that
$$ lim_xto x_0 f(x) = 0$$
for all $x_0 in [alpha, beta]$. Then $f(x) = 0$ for some $xin [alpha, beta]$.
The Thomae's function $f: [0,1]to mathbb R$
$$f(x) =begincases 1/q & textif x= p/qin mathbb Q, \ 0 & textotherwise.endcases$$
leads me to the above question. The Thomae function has limit zero everywhere, althought it is nonzero in $mathbb Q$. I think I can take any countable dense subset $Dsubset [alpha, beta]$ and construct a function which is nonzero in $D$ but limit equals zero everywhere. But I can't think of a function that is nonzero everywhere but has zero limit everywhere.
real-analysis limits
$endgroup$
$begingroup$
See math.stackexchange.com/q/1801935/72031 and math.stackexchange.com/q/980022/72031
$endgroup$
– Paramanand Singh
1 hour ago
add a comment |
$begingroup$
I wish to know if the following is true:
Let $f : [alpha, beta]to mathbb R$ be a function so that
$$ lim_xto x_0 f(x) = 0$$
for all $x_0 in [alpha, beta]$. Then $f(x) = 0$ for some $xin [alpha, beta]$.
The Thomae's function $f: [0,1]to mathbb R$
$$f(x) =begincases 1/q & textif x= p/qin mathbb Q, \ 0 & textotherwise.endcases$$
leads me to the above question. The Thomae function has limit zero everywhere, althought it is nonzero in $mathbb Q$. I think I can take any countable dense subset $Dsubset [alpha, beta]$ and construct a function which is nonzero in $D$ but limit equals zero everywhere. But I can't think of a function that is nonzero everywhere but has zero limit everywhere.
real-analysis limits
$endgroup$
I wish to know if the following is true:
Let $f : [alpha, beta]to mathbb R$ be a function so that
$$ lim_xto x_0 f(x) = 0$$
for all $x_0 in [alpha, beta]$. Then $f(x) = 0$ for some $xin [alpha, beta]$.
The Thomae's function $f: [0,1]to mathbb R$
$$f(x) =begincases 1/q & textif x= p/qin mathbb Q, \ 0 & textotherwise.endcases$$
leads me to the above question. The Thomae function has limit zero everywhere, althought it is nonzero in $mathbb Q$. I think I can take any countable dense subset $Dsubset [alpha, beta]$ and construct a function which is nonzero in $D$ but limit equals zero everywhere. But I can't think of a function that is nonzero everywhere but has zero limit everywhere.
real-analysis limits
real-analysis limits
asked 6 hours ago
Arctic CharArctic Char
157112
157112
$begingroup$
See math.stackexchange.com/q/1801935/72031 and math.stackexchange.com/q/980022/72031
$endgroup$
– Paramanand Singh
1 hour ago
add a comment |
$begingroup$
See math.stackexchange.com/q/1801935/72031 and math.stackexchange.com/q/980022/72031
$endgroup$
– Paramanand Singh
1 hour ago
$begingroup$
See math.stackexchange.com/q/1801935/72031 and math.stackexchange.com/q/980022/72031
$endgroup$
– Paramanand Singh
1 hour ago
$begingroup$
See math.stackexchange.com/q/1801935/72031 and math.stackexchange.com/q/980022/72031
$endgroup$
– Paramanand Singh
1 hour ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The number of points $xin[alpha,beta]$ at which $|f(x)|>1/n$, for a fixed $ninmathbbN$, has to be finite. Otherwise, since the interval is compact, they accumulate somewhere and at that point the limit wouldn't be zero.
Therefore, the points $xin[alpha,beta]$ at which $|f(x)|neq0$ is countable, but $[alpha,beta]$ isn't (unless $alpha=beta$ but that case follows directly).
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
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');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
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%2f3149783%2fa-limit-with-limit-zero-everywhere-must-be-zero-somewhere%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The number of points $xin[alpha,beta]$ at which $|f(x)|>1/n$, for a fixed $ninmathbbN$, has to be finite. Otherwise, since the interval is compact, they accumulate somewhere and at that point the limit wouldn't be zero.
Therefore, the points $xin[alpha,beta]$ at which $|f(x)|neq0$ is countable, but $[alpha,beta]$ isn't (unless $alpha=beta$ but that case follows directly).
$endgroup$
add a comment |
$begingroup$
The number of points $xin[alpha,beta]$ at which $|f(x)|>1/n$, for a fixed $ninmathbbN$, has to be finite. Otherwise, since the interval is compact, they accumulate somewhere and at that point the limit wouldn't be zero.
Therefore, the points $xin[alpha,beta]$ at which $|f(x)|neq0$ is countable, but $[alpha,beta]$ isn't (unless $alpha=beta$ but that case follows directly).
$endgroup$
add a comment |
$begingroup$
The number of points $xin[alpha,beta]$ at which $|f(x)|>1/n$, for a fixed $ninmathbbN$, has to be finite. Otherwise, since the interval is compact, they accumulate somewhere and at that point the limit wouldn't be zero.
Therefore, the points $xin[alpha,beta]$ at which $|f(x)|neq0$ is countable, but $[alpha,beta]$ isn't (unless $alpha=beta$ but that case follows directly).
$endgroup$
The number of points $xin[alpha,beta]$ at which $|f(x)|>1/n$, for a fixed $ninmathbbN$, has to be finite. Otherwise, since the interval is compact, they accumulate somewhere and at that point the limit wouldn't be zero.
Therefore, the points $xin[alpha,beta]$ at which $|f(x)|neq0$ is countable, but $[alpha,beta]$ isn't (unless $alpha=beta$ but that case follows directly).
answered 6 hours ago
user647486user647486
34615
34615
add a comment |
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');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
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%2f3149783%2fa-limit-with-limit-zero-everywhere-must-be-zero-somewhere%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');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
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');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
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');
var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);
;
$window.on('scroll', onScroll);
);
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
$begingroup$
See math.stackexchange.com/q/1801935/72031 and math.stackexchange.com/q/980022/72031
$endgroup$
– Paramanand Singh
1 hour ago