non negative distributional derivative

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Let $J$ be an open interval and let $f:JtomathbbR$ be locally integrable. Suppose that for all $0leqthetain C^infty_c(J)$ it holds
$$
int_J f(t),theta'(t),dtgeq 0,,
$$
namely the first distributional derivative of $f$ is non-negative.



Can one conclude that $f$ is non-increasing? Or does one need further assumptions?



The idea is the following: the equation tells us that actually $f'$ is not just a distribution but even a positive measure on $J$. How can I go on from this?






distribution-theory







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  • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    Sep 6 at 11:06










  • At least $f$ can be increasing on null sets.
    – md2perpe
    Sep 6 at 12:14










  • For example, consider $$f(x) = begincases -x & text if x in mathbbRsetminusmathbbQ,\ x & text if x in mathbbQ. endcases$$ Its distributional derivative is negative, but $f$ is not decreasing.
    – md2perpe
    Sep 6 at 14:31















up vote
0
down vote

favorite
1












Let $J$ be an open interval and let $f:JtomathbbR$ be locally integrable. Suppose that for all $0leqthetain C^infty_c(J)$ it holds
$$
int_J f(t),theta'(t),dtgeq 0,,
$$
namely the first distributional derivative of $f$ is non-negative.



Can one conclude that $f$ is non-increasing? Or does one need further assumptions?



The idea is the following: the equation tells us that actually $f'$ is not just a distribution but even a positive measure on $J$. How can I go on from this?






distribution-theory







share|cite|improve this question























  • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    Sep 6 at 11:06










  • At least $f$ can be increasing on null sets.
    – md2perpe
    Sep 6 at 12:14










  • For example, consider $$f(x) = begincases -x & text if x in mathbbRsetminusmathbbQ,\ x & text if x in mathbbQ. endcases$$ Its distributional derivative is negative, but $f$ is not decreasing.
    – md2perpe
    Sep 6 at 14:31













up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





Let $J$ be an open interval and let $f:JtomathbbR$ be locally integrable. Suppose that for all $0leqthetain C^infty_c(J)$ it holds
$$
int_J f(t),theta'(t),dtgeq 0,,
$$
namely the first distributional derivative of $f$ is non-negative.



Can one conclude that $f$ is non-increasing? Or does one need further assumptions?



The idea is the following: the equation tells us that actually $f'$ is not just a distribution but even a positive measure on $J$. How can I go on from this?






distribution-theory







share|cite|improve this question















Let $J$ be an open interval and let $f:JtomathbbR$ be locally integrable. Suppose that for all $0leqthetain C^infty_c(J)$ it holds
$$
int_J f(t),theta'(t),dtgeq 0,,
$$
namely the first distributional derivative of $f$ is non-negative.



Can one conclude that $f$ is non-increasing? Or does one need further assumptions?



The idea is the following: the equation tells us that actually $f'$ is not just a distribution but even a positive measure on $J$. How can I go on from this?






distribution-theory




distribution-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




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edited Sep 6 at 11:08

























asked Sep 6 at 11:03









user590533

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  • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    Sep 6 at 11:06










  • At least $f$ can be increasing on null sets.
    – md2perpe
    Sep 6 at 12:14










  • For example, consider $$f(x) = begincases -x & text if x in mathbbRsetminusmathbbQ,\ x & text if x in mathbbQ. endcases$$ Its distributional derivative is negative, but $f$ is not decreasing.
    – md2perpe
    Sep 6 at 14:31

















  • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    Sep 6 at 11:06










  • At least $f$ can be increasing on null sets.
    – md2perpe
    Sep 6 at 12:14










  • For example, consider $$f(x) = begincases -x & text if x in mathbbRsetminusmathbbQ,\ x & text if x in mathbbQ. endcases$$ Its distributional derivative is negative, but $f$ is not decreasing.
    – md2perpe
    Sep 6 at 14:31
















Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
– José Carlos Santos
Sep 6 at 11:06




Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
– José Carlos Santos
Sep 6 at 11:06












At least $f$ can be increasing on null sets.
– md2perpe
Sep 6 at 12:14




At least $f$ can be increasing on null sets.
– md2perpe
Sep 6 at 12:14












For example, consider $$f(x) = begincases -x & text if x in mathbbRsetminusmathbbQ,\ x & text if x in mathbbQ. endcases$$ Its distributional derivative is negative, but $f$ is not decreasing.
– md2perpe
Sep 6 at 14:31





For example, consider $$f(x) = begincases -x & text if x in mathbbRsetminusmathbbQ,\ x & text if x in mathbbQ. endcases$$ Its distributional derivative is negative, but $f$ is not decreasing.
– md2perpe
Sep 6 at 14:31
















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