Dtjytyk

Why doesn't the fusion process of the sun speed up?

Symbolism of 18 Journeyers

Why does Surtur say that Thor is Asgard's doom?

Hot air balloons as primitive bombers

label a part of commutative diagram

Have the tides ever turned twice on any open problem?

Are hand made posters acceptable in Academia?

Writing in a Christian voice

How do researchers send unsolicited emails asking for feedback on their works?

When should a starting writer get his own webpage?

Imaginary part of expression too difficult to calculate

Do I need to convey a moral for each of my blog post?

What is the reasoning behind standardization (dividing by standard deviation)?

Recursively updating the MLE as new observations stream in

Would mining huge amounts of resources on the Moon change its orbit?

What is it called when someone votes for an option that's not their first choice?

How can an organ that provides biological immortality be unable to regenerate?

Why are there no stars visible in cislunar space?

Nested Dynamic SOQL Query

Animating wave motion in water

Does the Shadow Magic sorcerer's Eyes of the Dark feature work on all Darkness spells or just his/her own?

Which partition to make active?

How to read string as hex number in bash?

Why is participating in the European Parliamentary elections used as a threat?

Prove or disprove sentence about $int f$

Prove or disprove three sentence about $int_{a}^{b}f(t)text{d}t$ (convexity)Limits of Monotone FunctionsClarification on wikipedia statement for discontinuities.Maximum and minimum values for a curveProve $2^ncdot n! ≤ (n+1)^n$ by induction.Does bounded variation imply boundednessCurve sketching for $ln(x^3 - x)$Understanding the notation $fcolon Dsubseteqmathbb{R}tomathbb{R}$ in the context of uniform continuityIntuition behind convolutionContinuous function with no local maximumProof explanation: Prove or disprove that $exists;K$ such that $|f(x)-f(y)|leq K|x-y|,;;forall; x,yin [0,1]$

1

$begingroup$

I post this question a few months ago. I solved the items (1) and (3), but I cannot to solve (2). Today I read this question again and I'm curious about solution of (2). Today, I had an idea, but I dont know if it works.

---

Idea. A monotone function has only jump discontinuities. So, $f|_{[f(x_{0}^{-}),f(x_{0}^{+})]}$ is continuous, then there is a maximum and minimum. If $w$ is a minimum on $[f(x_{0}^{-}),f(x_{0}^{+})]$, then

$$w leq f(x) Longrightarrow w(x-x_{0}) leq int_{x_{0}}^{x}f(t)dt = F(x) - F(x_{0})$$

taking, WLOG, $x geq x_{0}$. But, this works for $f$ on $[f(x_{0}^{-}),f(x_{0}^{+})]$. What about the general case?

real-analysis derivatives

share|cite|improve this question

asked Mar 12 at 5:30

Lucas CorrêaLucas Corrêa

1,4931421

$endgroup$

This question has an open bounty worth +50
reputation from Lucas Corrêa ending in 4 days.

This question has not received enough attention.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  $[f(x_0^+),f(x_0^-)]$ does not necessarily belong to the domain of $f$. You cannot define the restriction of $f$ on that interval.
  $endgroup$
  – nicomezi
  Mar 12 at 5:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @nicomezi I see. Do you have any hint? Or do you know if there is any way to adapt that idea?
  $endgroup$
  – Lucas Corrêa
  Mar 12 at 18:22
  

  
  
  
  

add a comment | 

1

$begingroup$

I post this question a few months ago. I solved the items (1) and (3), but I cannot to solve (2). Today I read this question again and I'm curious about solution of (2). Today, I had an idea, but I dont know if it works.

---

Idea. A monotone function has only jump discontinuities. So, $f|_{[f(x_{0}^{-}),f(x_{0}^{+})]}$ is continuous, then there is a maximum and minimum. If $w$ is a minimum on $[f(x_{0}^{-}),f(x_{0}^{+})]$, then

$$w leq f(x) Longrightarrow w(x-x_{0}) leq int_{x_{0}}^{x}f(t)dt = F(x) - F(x_{0})$$

taking, WLOG, $x geq x_{0}$. But, this works for $f$ on $[f(x_{0}^{-}),f(x_{0}^{+})]$. What about the general case?

real-analysis derivatives

share|cite|improve this question

asked Mar 12 at 5:30

Lucas CorrêaLucas Corrêa

1,4931421

$endgroup$

This question has an open bounty worth +50
reputation from Lucas Corrêa ending in 4 days.

This question has not received enough attention.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  $[f(x_0^+),f(x_0^-)]$ does not necessarily belong to the domain of $f$. You cannot define the restriction of $f$ on that interval.
  $endgroup$
  – nicomezi
  Mar 12 at 5:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @nicomezi I see. Do you have any hint? Or do you know if there is any way to adapt that idea?
  $endgroup$
  – Lucas Corrêa
  Mar 12 at 18:22
  

  
  
  
  

add a comment | 

$begingroup$

I post this question a few months ago. I solved the items (1) and (3), but I cannot to solve (2). Today I read this question again and I'm curious about solution of (2). Today, I had an idea, but I dont know if it works.

---

Idea. A monotone function has only jump discontinuities. So, $f|_{[f(x_{0}^{-}),f(x_{0}^{+})]}$ is continuous, then there is a maximum and minimum. If $w$ is a minimum on $[f(x_{0}^{-}),f(x_{0}^{+})]$, then

$$w leq f(x) Longrightarrow w(x-x_{0}) leq int_{x_{0}}^{x}f(t)dt = F(x) - F(x_{0})$$

taking, WLOG, $x geq x_{0}$. But, this works for $f$ on $[f(x_{0}^{-}),f(x_{0}^{+})]$. What about the general case?

real-analysis derivatives

share|cite|improve this question

asked Mar 12 at 5:30

Lucas CorrêaLucas Corrêa

1,4931421

$endgroup$

share|cite|improve this question

asked Mar 12 at 5:30

Lucas CorrêaLucas Corrêa

1,4931421

share|cite|improve this question

asked Mar 12 at 5:30

Lucas CorrêaLucas Corrêa

1,4931421

asked Mar 12 at 5:30

Lucas CorrêaLucas Corrêa

1,4931421

asked Mar 12 at 5:30

Lucas CorrêaLucas Corrêa

1,4931421

1 Answer
1

active

oldest

votes

1

$begingroup$

First suppose $x>x_0$. Then
$$F(x) - F(x_0) = int_{x_0}^{x}f(t)dt geq inf_{t in (x_0,x]} f(t) cdot (x-x_0) = f(x_0^+) cdot (x-x_0) geq w cdot (x-x_0).$$

Now let $x < x_0$. Then
$$F(x) - F(x_0) = int_{x_0}^{x}f(t)dt = - int_{x}^{x_0}f(t)dt = int_{x}^{x_0}(-f(t))dt geq inf_{t in [x,x_0)} (-f(t)) cdot (x_0-x)\
= (- sup_{t in [x,x_0)} f(t)) cdot (x_0-x) = sup_{t in [x,x_0)} f(t) cdot (x-x_0) = f(x_0^-) cdot (x-x_0) geq w cdot (x-x_0).$$

share|cite|improve this answer

answered Mar 15 at 20:01

Roman HricRoman Hric

1715

$endgroup$

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
});


}
});

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3144707%2fprove-or-disprove-sentence-about-int-f%23new-answer', 'question_page');
}
);

Post as a guest

Name

Email

Required, but never shown

1

$begingroup$

First suppose $x>x_0$. Then
$$F(x) - F(x_0) = int_{x_0}^{x}f(t)dt geq inf_{t in (x_0,x]} f(t) cdot (x-x_0) = f(x_0^+) cdot (x-x_0) geq w cdot (x-x_0).$$

Now let $x < x_0$. Then
$$F(x) - F(x_0) = int_{x_0}^{x}f(t)dt = - int_{x}^{x_0}f(t)dt = int_{x}^{x_0}(-f(t))dt geq inf_{t in [x,x_0)} (-f(t)) cdot (x_0-x)\
= (- sup_{t in [x,x_0)} f(t)) cdot (x_0-x) = sup_{t in [x,x_0)} f(t) cdot (x-x_0) = f(x_0^-) cdot (x-x_0) geq w cdot (x-x_0).$$

share|cite|improve this answer

answered Mar 15 at 20:01

Roman HricRoman Hric

1715

$endgroup$

add a comment | 

1

$begingroup$

First suppose $x>x_0$. Then
$$F(x) - F(x_0) = int_{x_0}^{x}f(t)dt geq inf_{t in (x_0,x]} f(t) cdot (x-x_0) = f(x_0^+) cdot (x-x_0) geq w cdot (x-x_0).$$

Now let $x < x_0$. Then
$$F(x) - F(x_0) = int_{x_0}^{x}f(t)dt = - int_{x}^{x_0}f(t)dt = int_{x}^{x_0}(-f(t))dt geq inf_{t in [x,x_0)} (-f(t)) cdot (x_0-x)\
= (- sup_{t in [x,x_0)} f(t)) cdot (x_0-x) = sup_{t in [x,x_0)} f(t) cdot (x-x_0) = f(x_0^-) cdot (x-x_0) geq w cdot (x-x_0).$$

share|cite|improve this answer

answered Mar 15 at 20:01

Roman HricRoman Hric

1715

$endgroup$

add a comment | 

$begingroup$

First suppose $x>x_0$. Then
$$F(x) - F(x_0) = int_{x_0}^{x}f(t)dt geq inf_{t in (x_0,x]} f(t) cdot (x-x_0) = f(x_0^+) cdot (x-x_0) geq w cdot (x-x_0).$$

Now let $x < x_0$. Then
$$F(x) - F(x_0) = int_{x_0}^{x}f(t)dt = - int_{x}^{x_0}f(t)dt = int_{x}^{x_0}(-f(t))dt geq inf_{t in [x,x_0)} (-f(t)) cdot (x_0-x)\
= (- sup_{t in [x,x_0)} f(t)) cdot (x_0-x) = sup_{t in [x,x_0)} f(t) cdot (x-x_0) = f(x_0^-) cdot (x-x_0) geq w cdot (x-x_0).$$

share|cite|improve this answer

answered Mar 15 at 20:01

Roman HricRoman Hric

1715

$endgroup$

share|cite|improve this answer

answered Mar 15 at 20:01

Roman HricRoman Hric

1715

answered Mar 15 at 20:01

Roman HricRoman Hric

1715

answered Mar 15 at 20:01

Roman HricRoman Hric

1715