If $F(x)$ is continuously differentiable and its derivatives are bounded, then $F(X)$ is absolutely...
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I know a result from measure theory saying that If $F(x)$ is differentiable at almost all $xin [a,b]$, its derivative $f=F'$ is in $L^1[a,b]$, and $F(b)-F(a) = int_{[a,b]}f(x),dx$, then $F$ is absolutely continuous.
I think that if $F(x)$ is differentiable and its derivative is bounded, then the first and second condition is satisfied. Also, $f(x) = F'(x)$ is continuous, and thus integrable. Hence, the third condition is also satisfied. Therefore, if $F(x)$ is continuously differentiable and its derivatives are bounded, then $F(X)$ is absolutely continuous.
Am I correct?
calculus measure-theory
asked Jan 23 at 15:48
user1292919user1292919
770512
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add a comment |
$begingroup$
I know a result from measure theory saying that If $F(x)$ is differentiable at almost all $xin [a,b]$, its derivative $f=F'$ is in $L^1[a,b]$, and $F(b)-F(a) = int_{[a,b]}f(x),dx$, then $F$ is absolutely continuous.
I think that if $F(x)$ is differentiable and its derivative is bounded, then the first and second condition is satisfied. Also, $f(x) = F'(x)$ is continuous, and thus integrable. Hence, the third condition is also satisfied. Therefore, if $F(x)$ is continuously differentiable and its derivatives are bounded, then $F(X)$ is absolutely continuous.
Am I correct?
calculus measure-theory
asked Jan 23 at 15:48
user1292919user1292919
770512
$endgroup$
add a comment |
$begingroup$
I know a result from measure theory saying that If $F(x)$ is differentiable at almost all $xin [a,b]$, its derivative $f=F'$ is in $L^1[a,b]$, and $F(b)-F(a) = int_{[a,b]}f(x),dx$, then $F$ is absolutely continuous.
I think that if $F(x)$ is differentiable and its derivative is bounded, then the first and second condition is satisfied. Also, $f(x) = F'(x)$ is continuous, and thus integrable. Hence, the third condition is also satisfied. Therefore, if $F(x)$ is continuously differentiable and its derivatives are bounded, then $F(X)$ is absolutely continuous.
Am I correct?
calculus measure-theory
asked Jan 23 at 15:48
user1292919user1292919
770512
$endgroup$
I know a result from measure theory saying that If $F(x)$ is differentiable at almost all $xin [a,b]$, its derivative $f=F'$ is in $L^1[a,b]$, and $F(b)-F(a) = int_{[a,b]}f(x),dx$, then $F$ is absolutely continuous.
I think that if $F(x)$ is differentiable and its derivative is bounded, then the first and second condition is satisfied. Also, $f(x) = F'(x)$ is continuous, and thus integrable. Hence, the third condition is also satisfied. Therefore, if $F(x)$ is continuously differentiable and its derivatives are bounded, then $F(X)$ is absolutely continuous.
Am I correct?
calculus measure-theory
calculus measure-theory
asked Jan 23 at 15:48
user1292919user1292919
770512
asked Jan 23 at 15:48
user1292919user1292919
770512
asked Jan 23 at 15:48
user1292919user1292919
770512
asked Jan 23 at 15:48
user1292919user1292919
770512
asked Jan 23 at 15:48
user1292919user1292919
770512
770512
add a comment |
add a comment |
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