How to ensure when the derivative approaches zero, the function value approaches a constant?
$begingroup$
I asked a very similar question here. But now this is different. Suppose $f(t)$ is differentiable and $c$ is a finite constant, then the following statement looks correct, but in fact it is not:
begin{equation}
limlimits_{t to infty} f'(t) = 0 implies limlimits_{t to infty} f(t)=c
end{equation}
A counter-example is $f(t)=ln(t)$. Now the question is, what condition should be used for the above statement to be true?
real-analysis derivatives
edited Dec 18 '18 at 17:12
zhw.
73.6k43175
asked Dec 18 '18 at 15:59
winstonwinston
524218
$endgroup$
add a comment |
$begingroup$
I asked a very similar question here. But now this is different. Suppose $f(t)$ is differentiable and $c$ is a finite constant, then the following statement looks correct, but in fact it is not:
begin{equation}
limlimits_{t to infty} f'(t) = 0 implies limlimits_{t to infty} f(t)=c
end{equation}
A counter-example is $f(t)=ln(t)$. Now the question is, what condition should be used for the above statement to be true?
real-analysis derivatives
edited Dec 18 '18 at 17:12
zhw.
73.6k43175
asked Dec 18 '18 at 15:59
winstonwinston
524218
$endgroup$
add a comment |
$begingroup$
I asked a very similar question here. But now this is different. Suppose $f(t)$ is differentiable and $c$ is a finite constant, then the following statement looks correct, but in fact it is not:
begin{equation}
limlimits_{t to infty} f'(t) = 0 implies limlimits_{t to infty} f(t)=c
end{equation}
A counter-example is $f(t)=ln(t)$. Now the question is, what condition should be used for the above statement to be true?
real-analysis derivatives
edited Dec 18 '18 at 17:12
zhw.
73.6k43175
asked Dec 18 '18 at 15:59
winstonwinston
524218
$endgroup$
I asked a very similar question here. But now this is different. Suppose $f(t)$ is differentiable and $c$ is a finite constant, then the following statement looks correct, but in fact it is not:
begin{equation}
limlimits_{t to infty} f'(t) = 0 implies limlimits_{t to infty} f(t)=c
end{equation}
A counter-example is $f(t)=ln(t)$. Now the question is, what condition should be used for the above statement to be true?
real-analysis derivatives
real-analysis derivatives
edited Dec 18 '18 at 17:12
zhw.
73.6k43175
asked Dec 18 '18 at 15:59
winstonwinston
524218
edited Dec 18 '18 at 17:12
zhw.
73.6k43175
asked Dec 18 '18 at 15:59
winstonwinston
524218
edited Dec 18 '18 at 17:12
zhw.
73.6k43175
edited Dec 18 '18 at 17:12
zhw.
73.6k43175
edited Dec 18 '18 at 17:12
zhw.
73.6k43175
73.6k43175
asked Dec 18 '18 at 15:59
winstonwinston
524218
asked Dec 18 '18 at 15:59
winstonwinston
524218
asked Dec 18 '18 at 15:59
winstonwinston
524218
524218
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
An equivalent condition is that
$$
int_a^infty f'(t),dt
$$
converges as an improper Riemann integral for some $ainBbb R$. This happens for instance if $|f'(x)|le C,x^{-p}$ for some $Cge0$ and $p>1$.
answered Dec 18 '18 at 16:24
Julián AguirreJulián Aguirre
69.1k24096
$endgroup$
$begingroup$
Is there any proof of your result?
$endgroup$
– winston
Dec 18 '18 at 16:39
$begingroup$
$$f(x)=f(a)+int_a^xf'(t),dt,quad x>a.$$
$endgroup$
– Julián Aguirre
Dec 18 '18 at 16:40
$begingroup$
True if $f'$ is Riemann integrable on bounded intervals.
$endgroup$
– zhw.
Dec 18 '18 at 17:14
$begingroup$
Does it mean that $limlimits_{t to infty} f'(t)=0$ does not make any difference to the condition? Doesn't it somehow relax your condition in some way?
$endgroup$
– winston
Dec 18 '18 at 20:21
$begingroup$
If $lim_{xtoinfty}f'(x)$ exists, it must be $0$. But it may happen that the limit does not exist.
$endgroup$
– Julián Aguirre
Dec 19 '18 at 8:28
add a comment |
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
An equivalent condition is that
$$
int_a^infty f'(t),dt
$$
converges as an improper Riemann integral for some $ainBbb R$. This happens for instance if $|f'(x)|le C,x^{-p}$ for some $Cge0$ and $p>1$.
answered Dec 18 '18 at 16:24
Julián AguirreJulián Aguirre
69.1k24096
$endgroup$
$begingroup$
Is there any proof of your result?
$endgroup$
– winston
Dec 18 '18 at 16:39
$begingroup$
$$f(x)=f(a)+int_a^xf'(t),dt,quad x>a.$$
$endgroup$
– Julián Aguirre
Dec 18 '18 at 16:40
$begingroup$
True if $f'$ is Riemann integrable on bounded intervals.
$endgroup$
– zhw.
Dec 18 '18 at 17:14
$begingroup$
Does it mean that $limlimits_{t to infty} f'(t)=0$ does not make any difference to the condition? Doesn't it somehow relax your condition in some way?
$endgroup$
– winston
Dec 18 '18 at 20:21
$begingroup$
If $lim_{xtoinfty}f'(x)$ exists, it must be $0$. But it may happen that the limit does not exist.
$endgroup$
– Julián Aguirre
Dec 19 '18 at 8:28
add a comment |
$begingroup$
An equivalent condition is that
$$
int_a^infty f'(t),dt
$$
converges as an improper Riemann integral for some $ainBbb R$. This happens for instance if $|f'(x)|le C,x^{-p}$ for some $Cge0$ and $p>1$.
answered Dec 18 '18 at 16:24
Julián AguirreJulián Aguirre
69.1k24096
$endgroup$
$begingroup$
Is there any proof of your result?
$endgroup$
– winston
Dec 18 '18 at 16:39
$begingroup$
$$f(x)=f(a)+int_a^xf'(t),dt,quad x>a.$$
$endgroup$
– Julián Aguirre
Dec 18 '18 at 16:40
$begingroup$
True if $f'$ is Riemann integrable on bounded intervals.
$endgroup$
– zhw.
Dec 18 '18 at 17:14
$begingroup$
Does it mean that $limlimits_{t to infty} f'(t)=0$ does not make any difference to the condition? Doesn't it somehow relax your condition in some way?
$endgroup$
– winston
Dec 18 '18 at 20:21
$begingroup$
If $lim_{xtoinfty}f'(x)$ exists, it must be $0$. But it may happen that the limit does not exist.
$endgroup$
– Julián Aguirre
Dec 19 '18 at 8:28
add a comment |
$begingroup$
An equivalent condition is that
$$
int_a^infty f'(t),dt
$$
converges as an improper Riemann integral for some $ainBbb R$. This happens for instance if $|f'(x)|le C,x^{-p}$ for some $Cge0$ and $p>1$.
answered Dec 18 '18 at 16:24
Julián AguirreJulián Aguirre
69.1k24096
$endgroup$
An equivalent condition is that
$$
int_a^infty f'(t),dt
$$
converges as an improper Riemann integral for some $ainBbb R$. This happens for instance if $|f'(x)|le C,x^{-p}$ for some $Cge0$ and $p>1$.
answered Dec 18 '18 at 16:24
Julián AguirreJulián Aguirre
69.1k24096
answered Dec 18 '18 at 16:24
Julián AguirreJulián Aguirre
69.1k24096
answered Dec 18 '18 at 16:24
Julián AguirreJulián Aguirre
69.1k24096
answered Dec 18 '18 at 16:24
Julián AguirreJulián Aguirre
69.1k24096
69.1k24096
$begingroup$
Is there any proof of your result?
$endgroup$
– winston
Dec 18 '18 at 16:39
$begingroup$
$$f(x)=f(a)+int_a^xf'(t),dt,quad x>a.$$
$endgroup$
– Julián Aguirre
Dec 18 '18 at 16:40
$begingroup$
True if $f'$ is Riemann integrable on bounded intervals.
$endgroup$
– zhw.
Dec 18 '18 at 17:14
$begingroup$
Does it mean that $limlimits_{t to infty} f'(t)=0$ does not make any difference to the condition? Doesn't it somehow relax your condition in some way?
$endgroup$
– winston
Dec 18 '18 at 20:21
$begingroup$
If $lim_{xtoinfty}f'(x)$ exists, it must be $0$. But it may happen that the limit does not exist.
$endgroup$
– Julián Aguirre
Dec 19 '18 at 8:28
add a comment |
$begingroup$
Is there any proof of your result?
$endgroup$
– winston
Dec 18 '18 at 16:39
$begingroup$
$$f(x)=f(a)+int_a^xf'(t),dt,quad x>a.$$
$endgroup$
– Julián Aguirre
Dec 18 '18 at 16:40
$begingroup$
True if $f'$ is Riemann integrable on bounded intervals.
$endgroup$
– zhw.
Dec 18 '18 at 17:14
$begingroup$
Does it mean that $limlimits_{t to infty} f'(t)=0$ does not make any difference to the condition? Doesn't it somehow relax your condition in some way?
$endgroup$
– winston
Dec 18 '18 at 20:21
$begingroup$
If $lim_{xtoinfty}f'(x)$ exists, it must be $0$. But it may happen that the limit does not exist.
$endgroup$
– Julián Aguirre
Dec 19 '18 at 8:28
$begingroup$
Is there any proof of your result?
$endgroup$
– winston
Dec 18 '18 at 16:39
$begingroup$
Is there any proof of your result?
$endgroup$
– winston
Dec 18 '18 at 16:39
$begingroup$
$$f(x)=f(a)+int_a^xf'(t),dt,quad x>a.$$
$endgroup$
– Julián Aguirre
Dec 18 '18 at 16:40
$begingroup$
$$f(x)=f(a)+int_a^xf'(t),dt,quad x>a.$$
$endgroup$
– Julián Aguirre
Dec 18 '18 at 16:40
$begingroup$
True if $f'$ is Riemann integrable on bounded intervals.
$endgroup$
– zhw.
Dec 18 '18 at 17:14
$begingroup$
True if $f'$ is Riemann integrable on bounded intervals.
$endgroup$
– zhw.
Dec 18 '18 at 17:14
$begingroup$
Does it mean that $limlimits_{t to infty} f'(t)=0$ does not make any difference to the condition? Doesn't it somehow relax your condition in some way?
$endgroup$
– winston
Dec 18 '18 at 20:21
$begingroup$
Does it mean that $limlimits_{t to infty} f'(t)=0$ does not make any difference to the condition? Doesn't it somehow relax your condition in some way?
$endgroup$
– winston
Dec 18 '18 at 20:21
$begingroup$
If $lim_{xtoinfty}f'(x)$ exists, it must be $0$. But it may happen that the limit does not exist.
$endgroup$
– Julián Aguirre
Dec 19 '18 at 8:28
$begingroup$
If $lim_{xtoinfty}f'(x)$ exists, it must be $0$. But it may happen that the limit does not exist.
$endgroup$
– Julián Aguirre
Dec 19 '18 at 8:28
add a comment |
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