Conditions for defining the Riemann Integral of $f$ over $[a,b]$ as a sum of rectangles with “uniform”...
$begingroup$
My book goes over how to prove that every continuous function $f: D rightarrow mathbb{R}$ has an antiderivative, which we previously determined/defined is the same as the Riemann Integral. The proof uses the distance between adjacent points of a partition on a closed interval $[a,b]$ to represent the "base" of the rectangles whose areas we sum to form the definite integral.
In a remark, it mentions that if we wanted to use "uniform" partitions, i.e. for any two points in the partition $t_j$, $t_{j+1}in [a,b]$, $(t_{j+1} - t_j) = frac{b-a}{N}$ where $N$ is the number of points in the partition, we must also show that $f$ is uniformly continuous.
Intuitively, I can see how this makes sense for a function like $f(x) = sin(frac{1}{x})$, where as we approach 0 on either side, the rectangles become a worse and worse approximation of the area under the graph. Is this the correct reasoning or am i missing something?
real-analysis riemann-integration
asked Dec 20 '18 at 1:48
hiroshinhiroshin
666
$endgroup$
add a comment |
$begingroup$
My book goes over how to prove that every continuous function $f: D rightarrow mathbb{R}$ has an antiderivative, which we previously determined/defined is the same as the Riemann Integral. The proof uses the distance between adjacent points of a partition on a closed interval $[a,b]$ to represent the "base" of the rectangles whose areas we sum to form the definite integral.
In a remark, it mentions that if we wanted to use "uniform" partitions, i.e. for any two points in the partition $t_j$, $t_{j+1}in [a,b]$, $(t_{j+1} - t_j) = frac{b-a}{N}$ where $N$ is the number of points in the partition, we must also show that $f$ is uniformly continuous.
Intuitively, I can see how this makes sense for a function like $f(x) = sin(frac{1}{x})$, where as we approach 0 on either side, the rectangles become a worse and worse approximation of the area under the graph. Is this the correct reasoning or am i missing something?
real-analysis riemann-integration
asked Dec 20 '18 at 1:48
hiroshinhiroshin
666
$endgroup$
add a comment |
$begingroup$
My book goes over how to prove that every continuous function $f: D rightarrow mathbb{R}$ has an antiderivative, which we previously determined/defined is the same as the Riemann Integral. The proof uses the distance between adjacent points of a partition on a closed interval $[a,b]$ to represent the "base" of the rectangles whose areas we sum to form the definite integral.
In a remark, it mentions that if we wanted to use "uniform" partitions, i.e. for any two points in the partition $t_j$, $t_{j+1}in [a,b]$, $(t_{j+1} - t_j) = frac{b-a}{N}$ where $N$ is the number of points in the partition, we must also show that $f$ is uniformly continuous.
Intuitively, I can see how this makes sense for a function like $f(x) = sin(frac{1}{x})$, where as we approach 0 on either side, the rectangles become a worse and worse approximation of the area under the graph. Is this the correct reasoning or am i missing something?
real-analysis riemann-integration
asked Dec 20 '18 at 1:48
hiroshinhiroshin
666
$endgroup$
My book goes over how to prove that every continuous function $f: D rightarrow mathbb{R}$ has an antiderivative, which we previously determined/defined is the same as the Riemann Integral. The proof uses the distance between adjacent points of a partition on a closed interval $[a,b]$ to represent the "base" of the rectangles whose areas we sum to form the definite integral.
In a remark, it mentions that if we wanted to use "uniform" partitions, i.e. for any two points in the partition $t_j$, $t_{j+1}in [a,b]$, $(t_{j+1} - t_j) = frac{b-a}{N}$ where $N$ is the number of points in the partition, we must also show that $f$ is uniformly continuous.
Intuitively, I can see how this makes sense for a function like $f(x) = sin(frac{1}{x})$, where as we approach 0 on either side, the rectangles become a worse and worse approximation of the area under the graph. Is this the correct reasoning or am i missing something?
real-analysis riemann-integration
real-analysis riemann-integration
asked Dec 20 '18 at 1:48
hiroshinhiroshin
666
asked Dec 20 '18 at 1:48
hiroshinhiroshin
666
edited Dec 20 '18 at 1:57
hiroshin
asked Dec 20 '18 at 1:48
hiroshinhiroshin
666
asked Dec 20 '18 at 1:48
hiroshinhiroshin
666
asked Dec 20 '18 at 1:48
hiroshinhiroshin
666
666
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%2f3047061%2fconditions-for-defining-the-riemann-integral-of-f-over-a-b-as-a-sum-of-rec%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%2f3047061%2fconditions-for-defining-the-riemann-integral-of-f-over-a-b-as-a-sum-of-rec%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
