Question regarding the formulation of a proof using the mean value theorem for integrals
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Let $f:[a,b]rightarrowmathbbR$ be continuous, and suppose that $int_c^df=0$ for every interval $[c,d]subset[a,b]$. Show that $f(x)=0$ for every $xin[a,b]$.
I just need assistance writing a formal proof of this. I believe I have the general idea:
By the mean value theorem, there exists an $zin[c,d]$ such that $f(z)(d-c)=0$. Because $int_c^df=0$ for $textitevery$ subinterval $[c,d]subset[a,b]$, it follows that $f(x)=0$ for every $xin[a,b]$.
How do I make this rigorous?
calculus real-analysis proof-verification definite-integrals
edited Aug 28 at 14:53
José Carlos Santos
120k16101182
asked Aug 28 at 14:49
Atsina
711115
add a comment |Â
up vote
0
down vote
favorite
Let $f:[a,b]rightarrowmathbbR$ be continuous, and suppose that $int_c^df=0$ for every interval $[c,d]subset[a,b]$. Show that $f(x)=0$ for every $xin[a,b]$.
I just need assistance writing a formal proof of this. I believe I have the general idea:
By the mean value theorem, there exists an $zin[c,d]$ such that $f(z)(d-c)=0$. Because $int_c^df=0$ for $textitevery$ subinterval $[c,d]subset[a,b]$, it follows that $f(x)=0$ for every $xin[a,b]$.
How do I make this rigorous?
calculus real-analysis proof-verification definite-integrals
edited Aug 28 at 14:53
José Carlos Santos
120k16101182
asked Aug 28 at 14:49
Atsina
711115
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $f:[a,b]rightarrowmathbbR$ be continuous, and suppose that $int_c^df=0$ for every interval $[c,d]subset[a,b]$. Show that $f(x)=0$ for every $xin[a,b]$.
I just need assistance writing a formal proof of this. I believe I have the general idea:
By the mean value theorem, there exists an $zin[c,d]$ such that $f(z)(d-c)=0$. Because $int_c^df=0$ for $textitevery$ subinterval $[c,d]subset[a,b]$, it follows that $f(x)=0$ for every $xin[a,b]$.
How do I make this rigorous?
calculus real-analysis proof-verification definite-integrals
edited Aug 28 at 14:53
José Carlos Santos
120k16101182
asked Aug 28 at 14:49
Atsina
711115
Let $f:[a,b]rightarrowmathbbR$ be continuous, and suppose that $int_c^df=0$ for every interval $[c,d]subset[a,b]$. Show that $f(x)=0$ for every $xin[a,b]$.
I just need assistance writing a formal proof of this. I believe I have the general idea:
By the mean value theorem, there exists an $zin[c,d]$ such that $f(z)(d-c)=0$. Because $int_c^df=0$ for $textitevery$ subinterval $[c,d]subset[a,b]$, it follows that $f(x)=0$ for every $xin[a,b]$.
How do I make this rigorous?
calculus real-analysis proof-verification definite-integrals
edited Aug 28 at 14:53
José Carlos Santos
120k16101182
asked Aug 28 at 14:49
Atsina
711115
edited Aug 28 at 14:53
José Carlos Santos
120k16101182
edited Aug 28 at 14:53
José Carlos Santos
120k16101182
edited Aug 28 at 14:53
José Carlos Santos
120k16101182
120k16101182
asked Aug 28 at 14:49
Atsina
711115
asked Aug 28 at 14:49
Atsina
711115
asked Aug 28 at 14:49
Atsina
711115
711115
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
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accepted
You proved correctly that every interval $[c,d]subset[a,b]$ (with more than one point) has a zero of $f$. Therefore, $xin[a,b],$ is dense in $[a,b]$. Since $f$ is countinuous, it follows from this that $f$ is the null function.
answered Aug 28 at 14:53
José Carlos Santos
120k16101182
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
You proved correctly that every interval $[c,d]subset[a,b]$ (with more than one point) has a zero of $f$. Therefore, $xin[a,b],$ is dense in $[a,b]$. Since $f$ is countinuous, it follows from this that $f$ is the null function.
answered Aug 28 at 14:53
José Carlos Santos
120k16101182
add a comment |Â
up vote
0
down vote
accepted
You proved correctly that every interval $[c,d]subset[a,b]$ (with more than one point) has a zero of $f$. Therefore, $xin[a,b],$ is dense in $[a,b]$. Since $f$ is countinuous, it follows from this that $f$ is the null function.
answered Aug 28 at 14:53
José Carlos Santos
120k16101182
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
You proved correctly that every interval $[c,d]subset[a,b]$ (with more than one point) has a zero of $f$. Therefore, $xin[a,b],$ is dense in $[a,b]$. Since $f$ is countinuous, it follows from this that $f$ is the null function.
answered Aug 28 at 14:53
José Carlos Santos
120k16101182
You proved correctly that every interval $[c,d]subset[a,b]$ (with more than one point) has a zero of $f$. Therefore, $xin[a,b],$ is dense in $[a,b]$. Since $f$ is countinuous, it follows from this that $f$ is the null function.
answered Aug 28 at 14:53
José Carlos Santos
120k16101182
answered Aug 28 at 14:53
José Carlos Santos
120k16101182
answered Aug 28 at 14:53
José Carlos Santos
120k16101182
answered Aug 28 at 14:53
José Carlos Santos
120k16101182
120k16101182
add a comment |Â
add a comment |Â
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