Is this problem right?
Let $f$ be a continuous function from an open interval $Isubseteqmathbb{R}$ to a Banach space $E$. The problem/exercise asks to prove that $f$ is differentiable in $x_0in I$ if and only if the limit
$$lim_{(h,k)to (0,0)}frac{f(x_0+h)-f(x_0-h)}{h+k}$$
exists when $h,kgt 0$.
Shouldn't it be
$$frac{f(x_0+h)-f(x_0-k)}{h+k}$$
otherwise i don't see what is $k$ doing there in the denominator. I've been able to do the exercise asuming it wanted to say "$k$" there because, if not, then $0=lim_{kto0^+}(lim_{hto0^+})$ can be very different to $lim_{hto0^+}(lim_{kto0^+})$ and they should be equal. Should I try it as it is?? The book this is from doesn't mess around and the $h$ could be on purpose to make the exercise very difficult.
Thanks
limits derivatives
add a comment |
Let $f$ be a continuous function from an open interval $Isubseteqmathbb{R}$ to a Banach space $E$. The problem/exercise asks to prove that $f$ is differentiable in $x_0in I$ if and only if the limit
$$lim_{(h,k)to (0,0)}frac{f(x_0+h)-f(x_0-h)}{h+k}$$
exists when $h,kgt 0$.
Shouldn't it be
$$frac{f(x_0+h)-f(x_0-k)}{h+k}$$
otherwise i don't see what is $k$ doing there in the denominator. I've been able to do the exercise asuming it wanted to say "$k$" there because, if not, then $0=lim_{kto0^+}(lim_{hto0^+})$ can be very different to $lim_{hto0^+}(lim_{kto0^+})$ and they should be equal. Should I try it as it is?? The book this is from doesn't mess around and the $h$ could be on purpose to make the exercise very difficult.
Thanks
limits derivatives
1
Surely, there is a typo in the exercise.
– Kavi Rama Murthy
Jan 3 at 23:24
add a comment |
Let $f$ be a continuous function from an open interval $Isubseteqmathbb{R}$ to a Banach space $E$. The problem/exercise asks to prove that $f$ is differentiable in $x_0in I$ if and only if the limit
$$lim_{(h,k)to (0,0)}frac{f(x_0+h)-f(x_0-h)}{h+k}$$
exists when $h,kgt 0$.
Shouldn't it be
$$frac{f(x_0+h)-f(x_0-k)}{h+k}$$
otherwise i don't see what is $k$ doing there in the denominator. I've been able to do the exercise asuming it wanted to say "$k$" there because, if not, then $0=lim_{kto0^+}(lim_{hto0^+})$ can be very different to $lim_{hto0^+}(lim_{kto0^+})$ and they should be equal. Should I try it as it is?? The book this is from doesn't mess around and the $h$ could be on purpose to make the exercise very difficult.
Thanks
limits derivatives
Let $f$ be a continuous function from an open interval $Isubseteqmathbb{R}$ to a Banach space $E$. The problem/exercise asks to prove that $f$ is differentiable in $x_0in I$ if and only if the limit
$$lim_{(h,k)to (0,0)}frac{f(x_0+h)-f(x_0-h)}{h+k}$$
exists when $h,kgt 0$.
Shouldn't it be
$$frac{f(x_0+h)-f(x_0-k)}{h+k}$$
otherwise i don't see what is $k$ doing there in the denominator. I've been able to do the exercise asuming it wanted to say "$k$" there because, if not, then $0=lim_{kto0^+}(lim_{hto0^+})$ can be very different to $lim_{hto0^+}(lim_{kto0^+})$ and they should be equal. Should I try it as it is?? The book this is from doesn't mess around and the $h$ could be on purpose to make the exercise very difficult.
Thanks
limits derivatives
limits derivatives
asked Jan 3 at 23:21
Pedro
517212
517212
1
Surely, there is a typo in the exercise.
– Kavi Rama Murthy
Jan 3 at 23:24
add a comment |
1
Surely, there is a typo in the exercise.
– Kavi Rama Murthy
Jan 3 at 23:24
1
1
Surely, there is a typo in the exercise.
– Kavi Rama Murthy
Jan 3 at 23:24
Surely, there is a typo in the exercise.
– Kavi Rama Murthy
Jan 3 at 23:24
add a comment |
1 Answer
1
active
oldest
votes
You are right. It is obviously a typo.
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%2f3061139%2fis-this-problem-right%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
You are right. It is obviously a typo.
add a comment |
You are right. It is obviously a typo.
add a comment |
You are right. It is obviously a typo.
You are right. It is obviously a typo.
answered Jan 3 at 23:24
José Carlos Santos
152k22123225
152k22123225
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3061139%2fis-this-problem-right%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
1
Surely, there is a typo in the exercise.
– Kavi Rama Murthy
Jan 3 at 23:24