Theorem or Justification for the function with a given finite integral over an interval with the lowwest...
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In a paper I'm writing, I'm trying to find a formal theorem or other concise way of stating that in the space of continuous functions whose definite integral over the over the interval $,0 leq tleq t_f$ is equal to a given constant, positive value,
$int_0^{t_f},mathcal{f}(t)=C$ , the function $mathcal{f}(t)$ with the lowest maximum value is a constant. The principle seems self evident but for my purposes I would like to have a way to show this definitively. Because I'm currently going off of the self evidence of the thing, I could very well be wrong, so I would also like to have a definitive answer before proceeding in the paper but I've been struggling to find any information on this problem, or to demonstrate it myself.
calculus
asked Jan 11 at 18:44
MargaretLMargaretL
11
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add a comment |
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In a paper I'm writing, I'm trying to find a formal theorem or other concise way of stating that in the space of continuous functions whose definite integral over the over the interval $,0 leq tleq t_f$ is equal to a given constant, positive value,
$int_0^{t_f},mathcal{f}(t)=C$ , the function $mathcal{f}(t)$ with the lowest maximum value is a constant. The principle seems self evident but for my purposes I would like to have a way to show this definitively. Because I'm currently going off of the self evidence of the thing, I could very well be wrong, so I would also like to have a definitive answer before proceeding in the paper but I've been struggling to find any information on this problem, or to demonstrate it myself.
calculus
asked Jan 11 at 18:44
MargaretLMargaretL
11
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This is false if you allow discontinuous functions.
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– Randall
Jan 11 at 18:45
1
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You are correct! Thank you very much. I've edited the question to allow only for continuous functions.
$endgroup$
– MargaretL
Jan 11 at 18:49
add a comment |
$begingroup$
In a paper I'm writing, I'm trying to find a formal theorem or other concise way of stating that in the space of continuous functions whose definite integral over the over the interval $,0 leq tleq t_f$ is equal to a given constant, positive value,
$int_0^{t_f},mathcal{f}(t)=C$ , the function $mathcal{f}(t)$ with the lowest maximum value is a constant. The principle seems self evident but for my purposes I would like to have a way to show this definitively. Because I'm currently going off of the self evidence of the thing, I could very well be wrong, so I would also like to have a definitive answer before proceeding in the paper but I've been struggling to find any information on this problem, or to demonstrate it myself.
calculus
asked Jan 11 at 18:44
MargaretLMargaretL
11
$endgroup$
In a paper I'm writing, I'm trying to find a formal theorem or other concise way of stating that in the space of continuous functions whose definite integral over the over the interval $,0 leq tleq t_f$ is equal to a given constant, positive value,
$int_0^{t_f},mathcal{f}(t)=C$ , the function $mathcal{f}(t)$ with the lowest maximum value is a constant. The principle seems self evident but for my purposes I would like to have a way to show this definitively. Because I'm currently going off of the self evidence of the thing, I could very well be wrong, so I would also like to have a definitive answer before proceeding in the paper but I've been struggling to find any information on this problem, or to demonstrate it myself.
calculus
calculus
asked Jan 11 at 18:44
MargaretLMargaretL
11
asked Jan 11 at 18:44
MargaretLMargaretL
11
edited Jan 11 at 18:48
MargaretL
asked Jan 11 at 18:44
MargaretLMargaretL
11
asked Jan 11 at 18:44
MargaretLMargaretL
11
asked Jan 11 at 18:44
MargaretLMargaretL
11
11
$begingroup$
This is false if you allow discontinuous functions.
$endgroup$
– Randall
Jan 11 at 18:45
1
$begingroup$
You are correct! Thank you very much. I've edited the question to allow only for continuous functions.
$endgroup$
– MargaretL
Jan 11 at 18:49
add a comment |
$begingroup$
This is false if you allow discontinuous functions.
$endgroup$
– Randall
Jan 11 at 18:45
1
$begingroup$
You are correct! Thank you very much. I've edited the question to allow only for continuous functions.
$endgroup$
– MargaretL
Jan 11 at 18:49
$begingroup$
This is false if you allow discontinuous functions.
$endgroup$
– Randall
Jan 11 at 18:45
$begingroup$
This is false if you allow discontinuous functions.
$endgroup$
– Randall
Jan 11 at 18:45
1
1
$begingroup$
You are correct! Thank you very much. I've edited the question to allow only for continuous functions.
$endgroup$
– MargaretL
Jan 11 at 18:49
$begingroup$
You are correct! Thank you very much. I've edited the question to allow only for continuous functions.
$endgroup$
– MargaretL
Jan 11 at 18:49
add a comment |
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$begingroup$
This is false if you allow discontinuous functions.
$endgroup$
– Randall
Jan 11 at 18:45
1
$begingroup$
You are correct! Thank you very much. I've edited the question to allow only for continuous functions.
$endgroup$
– MargaretL
Jan 11 at 18:49