riemann integrable functions map to another one
$begingroup$
I have the following question to prove:
If $f_{1}, f_{2},...,f_{n}$ are Riemann integrable functions, and $phi$ a function from $R^{n}$ to $R$, where $R$ denotes the set of real numbers, such that $phi$ is monotone with respect to each of its coordinates, prove that $phi(f_1,...,f_n)$ is also Riemann integrable.
I had the following: approach $phi(f_1,...,f_n)$ with $phi(psi_1,...,psi_n)$ where $psi_{i}$ is a step function that controls $f_i$ however I have some trouble controlling the difference without any hypothesis regarding the linearity.
EDIT: I'm currently trying with $n = 1$ , aka $phi$ is a monotone function. One thing that I know is that it has countable discontinuity points.
Could i get some hints? THanks in advance
analysis
asked Dec 11 '18 at 21:53
J.DoeJ.Doe
62
$endgroup$
add a comment |
$begingroup$
I have the following question to prove:
If $f_{1}, f_{2},...,f_{n}$ are Riemann integrable functions, and $phi$ a function from $R^{n}$ to $R$, where $R$ denotes the set of real numbers, such that $phi$ is monotone with respect to each of its coordinates, prove that $phi(f_1,...,f_n)$ is also Riemann integrable.
I had the following: approach $phi(f_1,...,f_n)$ with $phi(psi_1,...,psi_n)$ where $psi_{i}$ is a step function that controls $f_i$ however I have some trouble controlling the difference without any hypothesis regarding the linearity.
EDIT: I'm currently trying with $n = 1$ , aka $phi$ is a monotone function. One thing that I know is that it has countable discontinuity points.
Could i get some hints? THanks in advance
analysis
asked Dec 11 '18 at 21:53
J.DoeJ.Doe
62
$endgroup$
add a comment |
$begingroup$
I have the following question to prove:
If $f_{1}, f_{2},...,f_{n}$ are Riemann integrable functions, and $phi$ a function from $R^{n}$ to $R$, where $R$ denotes the set of real numbers, such that $phi$ is monotone with respect to each of its coordinates, prove that $phi(f_1,...,f_n)$ is also Riemann integrable.
I had the following: approach $phi(f_1,...,f_n)$ with $phi(psi_1,...,psi_n)$ where $psi_{i}$ is a step function that controls $f_i$ however I have some trouble controlling the difference without any hypothesis regarding the linearity.
EDIT: I'm currently trying with $n = 1$ , aka $phi$ is a monotone function. One thing that I know is that it has countable discontinuity points.
Could i get some hints? THanks in advance
analysis
asked Dec 11 '18 at 21:53
J.DoeJ.Doe
62
$endgroup$
I have the following question to prove:
If $f_{1}, f_{2},...,f_{n}$ are Riemann integrable functions, and $phi$ a function from $R^{n}$ to $R$, where $R$ denotes the set of real numbers, such that $phi$ is monotone with respect to each of its coordinates, prove that $phi(f_1,...,f_n)$ is also Riemann integrable.
I had the following: approach $phi(f_1,...,f_n)$ with $phi(psi_1,...,psi_n)$ where $psi_{i}$ is a step function that controls $f_i$ however I have some trouble controlling the difference without any hypothesis regarding the linearity.
EDIT: I'm currently trying with $n = 1$ , aka $phi$ is a monotone function. One thing that I know is that it has countable discontinuity points.
Could i get some hints? THanks in advance
analysis
analysis
asked Dec 11 '18 at 21:53
J.DoeJ.Doe
62
asked Dec 11 '18 at 21:53
J.DoeJ.Doe
62
edited Dec 12 '18 at 13:31
J.Doe
asked Dec 11 '18 at 21:53
J.DoeJ.Doe
62
asked Dec 11 '18 at 21:53
J.DoeJ.Doe
62
asked Dec 11 '18 at 21:53
J.DoeJ.Doe
62
62
add a comment |
add a comment |
0
active
oldest
votes
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');
});
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%2f3035883%2friemann-integrable-functions-map-to-another-one%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3035883%2friemann-integrable-functions-map-to-another-one%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
