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If one of the Dini derivatives is bounded, then f is Lipschitz

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2

3

$begingroup$

If one of its Dini derivatives (say $D^+$) is bounded show that a function $f$ satisfies a Lipschitz condition,

Definition of the upper right Dini derivative: $$D^+f(x) =
limsup_hto 0^+ fracf(x+h) - f(x)h$$

This is a question appearing in Royden's Real Analysis textbook. (Edition 3)

[Edited: The poster evidently posted this because he could not answer it [it's false], did not find a counterexample [it's easy] and did not know what the correct version of the problem should have been had Royden had a more careful editor. The problem is interesting for these reasons.]

real-analysis measure-theory derivatives

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edited Nov 1 '15 at 6:35

Guest_000

asked Oct 30 '15 at 6:46

Guest_000Guest_000

291212

$endgroup$

add a comment | 

2

3

$begingroup$

If one of its Dini derivatives (say $D^+$) is bounded show that a function $f$ satisfies a Lipschitz condition,

Definition of the upper right Dini derivative: $$D^+f(x) =
limsup_hto 0^+ fracf(x+h) - f(x)h$$

This is a question appearing in Royden's Real Analysis textbook. (Edition 3)

[Edited: The poster evidently posted this because he could not answer it [it's false], did not find a counterexample [it's easy] and did not know what the correct version of the problem should have been had Royden had a more careful editor. The problem is interesting for these reasons.]

real-analysis measure-theory derivatives

share|cite|improve this question

edited Nov 1 '15 at 6:35

Guest_000

asked Oct 30 '15 at 6:46

Guest_000Guest_000

291212

$endgroup$

add a comment | 

$begingroup$

If one of its Dini derivatives (say $D^+$) is bounded show that a function $f$ satisfies a Lipschitz condition,

Definition of the upper right Dini derivative: $$D^+f(x) =
limsup_hto 0^+ fracf(x+h) - f(x)h$$

This is a question appearing in Royden's Real Analysis textbook. (Edition 3)

[Edited: The poster evidently posted this because he could not answer it [it's false], did not find a counterexample [it's easy] and did not know what the correct version of the problem should have been had Royden had a more careful editor. The problem is interesting for these reasons.]

real-analysis measure-theory derivatives

share|cite|improve this question

edited Nov 1 '15 at 6:35

Guest_000

asked Oct 30 '15 at 6:46

Guest_000Guest_000

291212

$endgroup$

share|cite|improve this question

edited Nov 1 '15 at 6:35

Guest_000

asked Oct 30 '15 at 6:46

Guest_000Guest_000

291212

share|cite|improve this question

edited Nov 1 '15 at 6:35

Guest_000

asked Oct 30 '15 at 6:46

Guest_000Guest_000

291212

asked Oct 30 '15 at 6:46

Guest_000Guest_000

291212

asked Oct 30 '15 at 6:46

Guest_000Guest_000

291212

1 Answer
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active

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votes

3

$begingroup$

I knew somebody would find a reason to distrust Royden--his text has survived way too long and none of us later textbook writers have been able to knock it out.

Take the function $f(x)=x$ for $-infty < x < 0$ and $f(x)=1+x$ for $0leq x < infty$. Then $D^+f(x) =1 $ at every point but no Lipschitz condition and not even continuous.

The result that Royden was going for was a theorem of Dini's from 1878.

U. Dini, Fondamenti per la teoretica delle funzione di variabli
reali, Pisa 1878.

But Dini assumed that $f$ is continuous!

There is a full account in S. Saks, Theory of the Integral (1937) p.~204.

P.S. Royden's first edition was from 1963. The third edition, which is the only one I have, is from 1988 and the mistake survives in that edition. Does anyone have a later edition where the problem was corrected?

share|cite|improve this answer

edited Oct 30 '15 at 16:38

answered Oct 30 '15 at 15:53

B. S. ThomsonB. S. Thomson

2,9351517

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3

$begingroup$

I knew somebody would find a reason to distrust Royden--his text has survived way too long and none of us later textbook writers have been able to knock it out.

Take the function $f(x)=x$ for $-infty < x < 0$ and $f(x)=1+x$ for $0leq x < infty$. Then $D^+f(x) =1 $ at every point but no Lipschitz condition and not even continuous.

The result that Royden was going for was a theorem of Dini's from 1878.

U. Dini, Fondamenti per la teoretica delle funzione di variabli
reali, Pisa 1878.

But Dini assumed that $f$ is continuous!

There is a full account in S. Saks, Theory of the Integral (1937) p.~204.

P.S. Royden's first edition was from 1963. The third edition, which is the only one I have, is from 1988 and the mistake survives in that edition. Does anyone have a later edition where the problem was corrected?

share|cite|improve this answer

edited Oct 30 '15 at 16:38

answered Oct 30 '15 at 15:53

B. S. ThomsonB. S. Thomson

2,9351517

$endgroup$

add a comment | 

3

$begingroup$

I knew somebody would find a reason to distrust Royden--his text has survived way too long and none of us later textbook writers have been able to knock it out.

Take the function $f(x)=x$ for $-infty < x < 0$ and $f(x)=1+x$ for $0leq x < infty$. Then $D^+f(x) =1 $ at every point but no Lipschitz condition and not even continuous.

The result that Royden was going for was a theorem of Dini's from 1878.

U. Dini, Fondamenti per la teoretica delle funzione di variabli
reali, Pisa 1878.

But Dini assumed that $f$ is continuous!

There is a full account in S. Saks, Theory of the Integral (1937) p.~204.

P.S. Royden's first edition was from 1963. The third edition, which is the only one I have, is from 1988 and the mistake survives in that edition. Does anyone have a later edition where the problem was corrected?

share|cite|improve this answer

edited Oct 30 '15 at 16:38

answered Oct 30 '15 at 15:53

B. S. ThomsonB. S. Thomson

2,9351517

$endgroup$

add a comment | 

$begingroup$

I knew somebody would find a reason to distrust Royden--his text has survived way too long and none of us later textbook writers have been able to knock it out.

Take the function $f(x)=x$ for $-infty < x < 0$ and $f(x)=1+x$ for $0leq x < infty$. Then $D^+f(x) =1 $ at every point but no Lipschitz condition and not even continuous.

The result that Royden was going for was a theorem of Dini's from 1878.

U. Dini, Fondamenti per la teoretica delle funzione di variabli
reali, Pisa 1878.

But Dini assumed that $f$ is continuous!

There is a full account in S. Saks, Theory of the Integral (1937) p.~204.

P.S. Royden's first edition was from 1963. The third edition, which is the only one I have, is from 1988 and the mistake survives in that edition. Does anyone have a later edition where the problem was corrected?

share|cite|improve this answer

edited Oct 30 '15 at 16:38

answered Oct 30 '15 at 15:53

B. S. ThomsonB. S. Thomson

2,9351517

$endgroup$

share|cite|improve this answer

edited Oct 30 '15 at 16:38

answered Oct 30 '15 at 15:53

B. S. ThomsonB. S. Thomson

2,9351517

answered Oct 30 '15 at 15:53

B. S. ThomsonB. S. Thomson

2,9351517

answered Oct 30 '15 at 15:53

B. S. ThomsonB. S. Thomson

2,9351517