Mixing
Prove that for every $epsilon>0$, there exists $cin (a,b)$ so that $f(x)(c−a)<epsilon$ for all $xin...
$begingroup$
Prove that for every $epsilon>0$, there exists $cin (a,b)$ so that $f(x)(c−a)<epsilon$ for all $xin [a,b]$.
I'm so confused about how to even start this question. I've tried setting $c = (a+b) / 2$, so that it is in the interval but I have no clue what I should even be trying to achieve to get prove this statement.
The question also originally states that $f$ is bounded on $[a, b]$ and that for any $c in (a, b)$, $f$ is integrable on $[c, b]$.
real-analysis integration
edited Feb 5 at 7:02
Brahadeesh
6,51642365
asked Feb 3 at 0:16
veloxvictoryveloxvictory
61
$endgroup$
add a comment |
$begingroup$
Prove that for every $epsilon>0$, there exists $cin (a,b)$ so that $f(x)(c−a)<epsilon$ for all $xin [a,b]$.
I'm so confused about how to even start this question. I've tried setting $c = (a+b) / 2$, so that it is in the interval but I have no clue what I should even be trying to achieve to get prove this statement.
The question also originally states that $f$ is bounded on $[a, b]$ and that for any $c in (a, b)$, $f$ is integrable on $[c, b]$.
real-analysis integration
edited Feb 5 at 7:02
Brahadeesh
6,51642365
asked Feb 3 at 0:16
veloxvictoryveloxvictory
61
$endgroup$
$begingroup$
I suppose $f$ is a function mapping $[a,b]$ into $mathbb R$, but which properties does it have?
$endgroup$
– Mars Plastic
Feb 3 at 0:19
$begingroup$
f is bounded on [a, b] and that for any c ∈ (a, b), f is integrable on [c, b].
$endgroup$
– veloxvictory
Feb 3 at 0:23
$begingroup$
@veloxvictory please edit your question so that the premises and question are in the body of the main text - it is illegible as written right now...
$endgroup$
– Mariah
Feb 3 at 0:41
$begingroup$
Probably a duplicate of this: math.stackexchange.com/questions/287540/…
$endgroup$
– Mars Plastic
Feb 4 at 23:53
add a comment |
$begingroup$
Prove that for every $epsilon>0$, there exists $cin (a,b)$ so that $f(x)(c−a)<epsilon$ for all $xin [a,b]$.
I'm so confused about how to even start this question. I've tried setting $c = (a+b) / 2$, so that it is in the interval but I have no clue what I should even be trying to achieve to get prove this statement.
The question also originally states that $f$ is bounded on $[a, b]$ and that for any $c in (a, b)$, $f$ is integrable on $[c, b]$.
real-analysis integration
edited Feb 5 at 7:02
Brahadeesh
6,51642365
asked Feb 3 at 0:16
veloxvictoryveloxvictory
61
$endgroup$
Prove that for every $epsilon>0$, there exists $cin (a,b)$ so that $f(x)(c−a)<epsilon$ for all $xin [a,b]$.
I'm so confused about how to even start this question. I've tried setting $c = (a+b) / 2$, so that it is in the interval but I have no clue what I should even be trying to achieve to get prove this statement.
The question also originally states that $f$ is bounded on $[a, b]$ and that for any $c in (a, b)$, $f$ is integrable on $[c, b]$.
real-analysis integration
real-analysis integration
edited Feb 5 at 7:02
Brahadeesh
6,51642365
asked Feb 3 at 0:16
veloxvictoryveloxvictory
61
edited Feb 5 at 7:02
Brahadeesh
6,51642365
asked Feb 3 at 0:16
veloxvictoryveloxvictory
61
edited Feb 5 at 7:02
Brahadeesh
6,51642365
edited Feb 5 at 7:02
Brahadeesh
6,51642365
edited Feb 5 at 7:02
Brahadeesh
6,51642365
6,51642365
asked Feb 3 at 0:16
veloxvictoryveloxvictory
61
asked Feb 3 at 0:16
veloxvictoryveloxvictory
61
asked Feb 3 at 0:16
veloxvictoryveloxvictory
61
61
$begingroup$
I suppose $f$ is a function mapping $[a,b]$ into $mathbb R$, but which properties does it have?
$endgroup$
– Mars Plastic
Feb 3 at 0:19
$begingroup$
f is bounded on [a, b] and that for any c ∈ (a, b), f is integrable on [c, b].
$endgroup$
– veloxvictory
Feb 3 at 0:23
$begingroup$
@veloxvictory please edit your question so that the premises and question are in the body of the main text - it is illegible as written right now...
$endgroup$
– Mariah
Feb 3 at 0:41
$begingroup$
Probably a duplicate of this: math.stackexchange.com/questions/287540/…
$endgroup$
– Mars Plastic
Feb 4 at 23:53
add a comment |
$begingroup$
I suppose $f$ is a function mapping $[a,b]$ into $mathbb R$, but which properties does it have?
$endgroup$
– Mars Plastic
Feb 3 at 0:19
$begingroup$
f is bounded on [a, b] and that for any c ∈ (a, b), f is integrable on [c, b].
$endgroup$
– veloxvictory
Feb 3 at 0:23
$begingroup$
@veloxvictory please edit your question so that the premises and question are in the body of the main text - it is illegible as written right now...
$endgroup$
– Mariah
Feb 3 at 0:41
$begingroup$
Probably a duplicate of this: math.stackexchange.com/questions/287540/…
$endgroup$
– Mars Plastic
Feb 4 at 23:53
$begingroup$
I suppose $f$ is a function mapping $[a,b]$ into $mathbb R$, but which properties does it have?
$endgroup$
– Mars Plastic
Feb 3 at 0:19
$begingroup$
I suppose $f$ is a function mapping $[a,b]$ into $mathbb R$, but which properties does it have?
$endgroup$
– Mars Plastic
Feb 3 at 0:19
$begingroup$
f is bounded on [a, b] and that for any c ∈ (a, b), f is integrable on [c, b].
$endgroup$
– veloxvictory
Feb 3 at 0:23
$begingroup$
f is bounded on [a, b] and that for any c ∈ (a, b), f is integrable on [c, b].
$endgroup$
– veloxvictory
Feb 3 at 0:23
$begingroup$
@veloxvictory please edit your question so that the premises and question are in the body of the main text - it is illegible as written right now...
$endgroup$
– Mariah
Feb 3 at 0:41
$begingroup$
@veloxvictory please edit your question so that the premises and question are in the body of the main text - it is illegible as written right now...
$endgroup$
– Mariah
Feb 3 at 0:41
$begingroup$
Probably a duplicate of this: math.stackexchange.com/questions/287540/…
$endgroup$
– Mars Plastic
Feb 4 at 23:53
$begingroup$
Probably a duplicate of this: math.stackexchange.com/questions/287540/…
$endgroup$
– Mars Plastic
Feb 4 at 23:53
add a comment |
1 Answer
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$begingroup$
If $f$ is bounded, then there is some $Min (0,infty)$ such that $|f(x)|le M$ for all $xin[a,b]$. If you choose any $cin(a,a+epsilon/M)$, you get
$$ f(x)(c-a)le |f(x)|epsilon/Mle epsilon quad text{for all $xin[a,b]$.}$$
However, I doubt that this is the actual question, since this does not need integrability.
answered Feb 3 at 0:38
Mars PlasticMars Plastic
1,465122
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
If $f$ is bounded, then there is some $Min (0,infty)$ such that $|f(x)|le M$ for all $xin[a,b]$. If you choose any $cin(a,a+epsilon/M)$, you get
$$ f(x)(c-a)le |f(x)|epsilon/Mle epsilon quad text{for all $xin[a,b]$.}$$
However, I doubt that this is the actual question, since this does not need integrability.
answered Feb 3 at 0:38
Mars PlasticMars Plastic
1,465122
$endgroup$
add a comment |
$begingroup$
If $f$ is bounded, then there is some $Min (0,infty)$ such that $|f(x)|le M$ for all $xin[a,b]$. If you choose any $cin(a,a+epsilon/M)$, you get
$$ f(x)(c-a)le |f(x)|epsilon/Mle epsilon quad text{for all $xin[a,b]$.}$$
However, I doubt that this is the actual question, since this does not need integrability.
answered Feb 3 at 0:38
Mars PlasticMars Plastic
1,465122
$endgroup$
add a comment |
$begingroup$
If $f$ is bounded, then there is some $Min (0,infty)$ such that $|f(x)|le M$ for all $xin[a,b]$. If you choose any $cin(a,a+epsilon/M)$, you get
$$ f(x)(c-a)le |f(x)|epsilon/Mle epsilon quad text{for all $xin[a,b]$.}$$
However, I doubt that this is the actual question, since this does not need integrability.
answered Feb 3 at 0:38
Mars PlasticMars Plastic
1,465122
$endgroup$
If $f$ is bounded, then there is some $Min (0,infty)$ such that $|f(x)|le M$ for all $xin[a,b]$. If you choose any $cin(a,a+epsilon/M)$, you get
$$ f(x)(c-a)le |f(x)|epsilon/Mle epsilon quad text{for all $xin[a,b]$.}$$
However, I doubt that this is the actual question, since this does not need integrability.
answered Feb 3 at 0:38
Mars PlasticMars Plastic
1,465122
answered Feb 3 at 0:38
Mars PlasticMars Plastic
1,465122
answered Feb 3 at 0:38
Mars PlasticMars Plastic
1,465122
answered Feb 3 at 0:38
Mars PlasticMars Plastic
1,465122
1,465122
add a comment |
add a comment |
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$begingroup$
I suppose $f$ is a function mapping $[a,b]$ into $mathbb R$, but which properties does it have?
$endgroup$
– Mars Plastic
Feb 3 at 0:19
$begingroup$
f is bounded on [a, b] and that for any c ∈ (a, b), f is integrable on [c, b].
$endgroup$
– veloxvictory
Feb 3 at 0:23
$begingroup$
@veloxvictory please edit your question so that the premises and question are in the body of the main text - it is illegible as written right now...
$endgroup$
– Mariah
Feb 3 at 0:41
$begingroup$
Probably a duplicate of this: math.stackexchange.com/questions/287540/…
$endgroup$
– Mars Plastic
Feb 4 at 23:53