Give an example of a continuous function that is not differentiable on the immediate left of 0
$begingroup$
Can we find a continuous function f: R→R such that f is not differentiable on (-δ, 0) for sufficiently small positive δ?
When I first think about it, I mistakenly think of the left differentiability of the function at x = 0, which is a different concept, and so come up with wrong examples such as
$$
f(x) = left{
begin{array}{ll}
sinBig(frac{1}{x}Big) & xneq 0 \
0 & x = 0\
end{array}
right.
$$
I think the Weierstrass' function works, but I am looking for something simpler (i.e. closed-form expressions). All elementary functions I know are only non-differentiable at "isolated" points, but not on an interval, so I cannot think of an appropriate example. Thank you in advance!
calculus derivatives continuity
asked Dec 27 '18 at 15:44
John LeiJohn Lei
445
$endgroup$
add a comment |
$begingroup$
Can we find a continuous function f: R→R such that f is not differentiable on (-δ, 0) for sufficiently small positive δ?
When I first think about it, I mistakenly think of the left differentiability of the function at x = 0, which is a different concept, and so come up with wrong examples such as
$$
f(x) = left{
begin{array}{ll}
sinBig(frac{1}{x}Big) & xneq 0 \
0 & x = 0\
end{array}
right.
$$
I think the Weierstrass' function works, but I am looking for something simpler (i.e. closed-form expressions). All elementary functions I know are only non-differentiable at "isolated" points, but not on an interval, so I cannot think of an appropriate example. Thank you in advance!
calculus derivatives continuity
asked Dec 27 '18 at 15:44
John LeiJohn Lei
445
$endgroup$
$begingroup$
Any function not defined on the negatives will trivially work, like $;sqrt x,,,log x;$ and etc. A function defined on a point and non-differentiable there is also to come upwith, but a function non-differentiable and defined on a complete interval is way more difficult to come up with
$endgroup$
– DonAntonio
Dec 27 '18 at 16:02
add a comment |
$begingroup$
Can we find a continuous function f: R→R such that f is not differentiable on (-δ, 0) for sufficiently small positive δ?
When I first think about it, I mistakenly think of the left differentiability of the function at x = 0, which is a different concept, and so come up with wrong examples such as
$$
f(x) = left{
begin{array}{ll}
sinBig(frac{1}{x}Big) & xneq 0 \
0 & x = 0\
end{array}
right.
$$
I think the Weierstrass' function works, but I am looking for something simpler (i.e. closed-form expressions). All elementary functions I know are only non-differentiable at "isolated" points, but not on an interval, so I cannot think of an appropriate example. Thank you in advance!
calculus derivatives continuity
asked Dec 27 '18 at 15:44
John LeiJohn Lei
445
$endgroup$
Can we find a continuous function f: R→R such that f is not differentiable on (-δ, 0) for sufficiently small positive δ?
When I first think about it, I mistakenly think of the left differentiability of the function at x = 0, which is a different concept, and so come up with wrong examples such as
$$
f(x) = left{
begin{array}{ll}
sinBig(frac{1}{x}Big) & xneq 0 \
0 & x = 0\
end{array}
right.
$$
I think the Weierstrass' function works, but I am looking for something simpler (i.e. closed-form expressions). All elementary functions I know are only non-differentiable at "isolated" points, but not on an interval, so I cannot think of an appropriate example. Thank you in advance!
calculus derivatives continuity
calculus derivatives continuity
asked Dec 27 '18 at 15:44
John LeiJohn Lei
445
asked Dec 27 '18 at 15:44
John LeiJohn Lei
445
asked Dec 27 '18 at 15:44
John LeiJohn Lei
445
asked Dec 27 '18 at 15:44
John LeiJohn Lei
445
asked Dec 27 '18 at 15:44
John LeiJohn Lei
445
445
$begingroup$
Any function not defined on the negatives will trivially work, like $;sqrt x,,,log x;$ and etc. A function defined on a point and non-differentiable there is also to come upwith, but a function non-differentiable and defined on a complete interval is way more difficult to come up with
$endgroup$
– DonAntonio
Dec 27 '18 at 16:02
add a comment |
$begingroup$
Any function not defined on the negatives will trivially work, like $;sqrt x,,,log x;$ and etc. A function defined on a point and non-differentiable there is also to come upwith, but a function non-differentiable and defined on a complete interval is way more difficult to come up with
$endgroup$
– DonAntonio
Dec 27 '18 at 16:02
$begingroup$
Any function not defined on the negatives will trivially work, like $;sqrt x,,,log x;$ and etc. A function defined on a point and non-differentiable there is also to come upwith, but a function non-differentiable and defined on a complete interval is way more difficult to come up with
$endgroup$
– DonAntonio
Dec 27 '18 at 16:02
$begingroup$
Any function not defined on the negatives will trivially work, like $;sqrt x,,,log x;$ and etc. A function defined on a point and non-differentiable there is also to come upwith, but a function non-differentiable and defined on a complete interval is way more difficult to come up with
$endgroup$
– DonAntonio
Dec 27 '18 at 16:02
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
For as far as I know, all elementary functions are differentiable almost everywhere in their domain. Even adding the possibility of integrating and adding interval piecewise definitions will still result in functions differentiable almost everywhere in their domain. I expect the only way to define a continuous function not differentiable in a set of positive measure is by taking limits (or infinite sums), but then you better just refer to the Weierstrass function.
answered Dec 27 '18 at 16:00
SmileyCraftSmileyCraft
3,591517
$endgroup$
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');
});
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%2f3054068%2fgive-an-example-of-a-continuous-function-that-is-not-differentiable-on-the-immed%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
$begingroup$
For as far as I know, all elementary functions are differentiable almost everywhere in their domain. Even adding the possibility of integrating and adding interval piecewise definitions will still result in functions differentiable almost everywhere in their domain. I expect the only way to define a continuous function not differentiable in a set of positive measure is by taking limits (or infinite sums), but then you better just refer to the Weierstrass function.
answered Dec 27 '18 at 16:00
SmileyCraftSmileyCraft
3,591517
$endgroup$
add a comment |
$begingroup$
For as far as I know, all elementary functions are differentiable almost everywhere in their domain. Even adding the possibility of integrating and adding interval piecewise definitions will still result in functions differentiable almost everywhere in their domain. I expect the only way to define a continuous function not differentiable in a set of positive measure is by taking limits (or infinite sums), but then you better just refer to the Weierstrass function.
answered Dec 27 '18 at 16:00
SmileyCraftSmileyCraft
3,591517
$endgroup$
add a comment |
$begingroup$
For as far as I know, all elementary functions are differentiable almost everywhere in their domain. Even adding the possibility of integrating and adding interval piecewise definitions will still result in functions differentiable almost everywhere in their domain. I expect the only way to define a continuous function not differentiable in a set of positive measure is by taking limits (or infinite sums), but then you better just refer to the Weierstrass function.
answered Dec 27 '18 at 16:00
SmileyCraftSmileyCraft
3,591517
$endgroup$
For as far as I know, all elementary functions are differentiable almost everywhere in their domain. Even adding the possibility of integrating and adding interval piecewise definitions will still result in functions differentiable almost everywhere in their domain. I expect the only way to define a continuous function not differentiable in a set of positive measure is by taking limits (or infinite sums), but then you better just refer to the Weierstrass function.
answered Dec 27 '18 at 16:00
SmileyCraftSmileyCraft
3,591517
answered Dec 27 '18 at 16:00
SmileyCraftSmileyCraft
3,591517
answered Dec 27 '18 at 16:00
SmileyCraftSmileyCraft
3,591517
answered Dec 27 '18 at 16:00
SmileyCraftSmileyCraft
3,591517
3,591517
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');
});
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%2f3054068%2fgive-an-example-of-a-continuous-function-that-is-not-differentiable-on-the-immed%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
$begingroup$
Any function not defined on the negatives will trivially work, like $;sqrt x,,,log x;$ and etc. A function defined on a point and non-differentiable there is also to come upwith, but a function non-differentiable and defined on a complete interval is way more difficult to come up with
$endgroup$
– DonAntonio
Dec 27 '18 at 16:02