Studying the total variation of an Integral Function

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Yesterday I was dealing with this problem:
Let $f : mathbbR-0 rightarrow mathbbR$ continuos and nonnegative and:



$$F(x)= int_1^x fracf(s)s ds hspace5mmx gt 0 $$
$$ F(x) = int_-1^x fracf(s)sds hspace5mm x lt 0$$



-For which $f$ we have $F in BV(mathbbR)$?



-For these $f$ give a formula for the total variation of $F$.



Taking the case $x>0$,
I was thinking that because of $f$ is continuos by the fundamental theorem of calculus it should be $F'(t) = fracf(t)t hspace3mm forall t in [1,x]$.
So $F'$ is continuous (because is equal to a composition of continuos functions) and we can use the formula:



$$T_F(x) = int_1^x |F'(t)| dt $$



At this point we must choose $f$ such that
$$int_1^+infty left|fracf(s)sright| ds lt infty$$



Maybe I'm totally wrong.




real-analysis analysis bounded-variation




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  • You got one quarter of the first question allright. For the second question we also need the value of $F(0)$.
    – Christian Blatter
    2 days ago










  • Can you be more specific? What do you mean by 'one quarter allright?😅 And also didn't understand properly why do we need the value F(0)?
    – James Arten
    2 days ago














up vote
2
down vote

favorite












Yesterday I was dealing with this problem:
Let $f : mathbbR-0 rightarrow mathbbR$ continuos and nonnegative and:



$$F(x)= int_1^x fracf(s)s ds hspace5mmx gt 0 $$
$$ F(x) = int_-1^x fracf(s)sds hspace5mm x lt 0$$



-For which $f$ we have $F in BV(mathbbR)$?



-For these $f$ give a formula for the total variation of $F$.



Taking the case $x>0$,
I was thinking that because of $f$ is continuos by the fundamental theorem of calculus it should be $F'(t) = fracf(t)t hspace3mm forall t in [1,x]$.
So $F'$ is continuous (because is equal to a composition of continuos functions) and we can use the formula:



$$T_F(x) = int_1^x |F'(t)| dt $$



At this point we must choose $f$ such that
$$int_1^+infty left|fracf(s)sright| ds lt infty$$



Maybe I'm totally wrong.




real-analysis analysis bounded-variation




share|cite|improve this question





















  • You got one quarter of the first question allright. For the second question we also need the value of $F(0)$.
    – Christian Blatter
    2 days ago










  • Can you be more specific? What do you mean by 'one quarter allright?😅 And also didn't understand properly why do we need the value F(0)?
    – James Arten
    2 days ago












up vote
2
down vote

favorite









up vote
2
down vote

favorite











Yesterday I was dealing with this problem:
Let $f : mathbbR-0 rightarrow mathbbR$ continuos and nonnegative and:



$$F(x)= int_1^x fracf(s)s ds hspace5mmx gt 0 $$
$$ F(x) = int_-1^x fracf(s)sds hspace5mm x lt 0$$



-For which $f$ we have $F in BV(mathbbR)$?



-For these $f$ give a formula for the total variation of $F$.



Taking the case $x>0$,
I was thinking that because of $f$ is continuos by the fundamental theorem of calculus it should be $F'(t) = fracf(t)t hspace3mm forall t in [1,x]$.
So $F'$ is continuous (because is equal to a composition of continuos functions) and we can use the formula:



$$T_F(x) = int_1^x |F'(t)| dt $$



At this point we must choose $f$ such that
$$int_1^+infty left|fracf(s)sright| ds lt infty$$



Maybe I'm totally wrong.




real-analysis analysis bounded-variation




share|cite|improve this question













Yesterday I was dealing with this problem:
Let $f : mathbbR-0 rightarrow mathbbR$ continuos and nonnegative and:



$$F(x)= int_1^x fracf(s)s ds hspace5mmx gt 0 $$
$$ F(x) = int_-1^x fracf(s)sds hspace5mm x lt 0$$



-For which $f$ we have $F in BV(mathbbR)$?



-For these $f$ give a formula for the total variation of $F$.



Taking the case $x>0$,
I was thinking that because of $f$ is continuos by the fundamental theorem of calculus it should be $F'(t) = fracf(t)t hspace3mm forall t in [1,x]$.
So $F'$ is continuous (because is equal to a composition of continuos functions) and we can use the formula:



$$T_F(x) = int_1^x |F'(t)| dt $$



At this point we must choose $f$ such that
$$int_1^+infty left|fracf(s)sright| ds lt infty$$



Maybe I'm totally wrong.





real-analysis analysis bounded-variation





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited 2 days ago
























asked 2 days ago









James Arten

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  • You got one quarter of the first question allright. For the second question we also need the value of $F(0)$.
    – Christian Blatter
    2 days ago










  • Can you be more specific? What do you mean by 'one quarter allright?😅 And also didn't understand properly why do we need the value F(0)?
    – James Arten
    2 days ago
















  • You got one quarter of the first question allright. For the second question we also need the value of $F(0)$.
    – Christian Blatter
    2 days ago










  • Can you be more specific? What do you mean by 'one quarter allright?😅 And also didn't understand properly why do we need the value F(0)?
    – James Arten
    2 days ago















You got one quarter of the first question allright. For the second question we also need the value of $F(0)$.
– Christian Blatter
2 days ago




You got one quarter of the first question allright. For the second question we also need the value of $F(0)$.
– Christian Blatter
2 days ago












Can you be more specific? What do you mean by 'one quarter allright?😅 And also didn't understand properly why do we need the value F(0)?
– James Arten
2 days ago




Can you be more specific? What do you mean by 'one quarter allright?😅 And also didn't understand properly why do we need the value F(0)?
– James Arten
2 days ago















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