Continuous and bounded functions and Riemann integrability
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Suppose that $v=v(t,x)in C^1([0,+infty]timesmathbb{R})$ is a compactly supported function. Is it true that $$v(0,x):mathbb{R}to mathbb{R}$$ is Riemann integrable over $mathbb{R}$?
Certainly $v(0,x)$ is a $C^1$ function and also it is bounded (in particular, $v(0,x)$ is zero outside a bounded subset of $mathbb{R}$). Is there any theorem that allows me to state that $v(0,x)$ is integrable over $mathbb{R}$?
Thanks a lot in advance.
real-analysis compactness riemann-integration
asked Nov 20 at 7:44
eleguitar
115114
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up vote
0
down vote
favorite
Suppose that $v=v(t,x)in C^1([0,+infty]timesmathbb{R})$ is a compactly supported function. Is it true that $$v(0,x):mathbb{R}to mathbb{R}$$ is Riemann integrable over $mathbb{R}$?
Certainly $v(0,x)$ is a $C^1$ function and also it is bounded (in particular, $v(0,x)$ is zero outside a bounded subset of $mathbb{R}$). Is there any theorem that allows me to state that $v(0,x)$ is integrable over $mathbb{R}$?
Thanks a lot in advance.
real-analysis compactness riemann-integration
asked Nov 20 at 7:44
eleguitar
115114
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose that $v=v(t,x)in C^1([0,+infty]timesmathbb{R})$ is a compactly supported function. Is it true that $$v(0,x):mathbb{R}to mathbb{R}$$ is Riemann integrable over $mathbb{R}$?
Certainly $v(0,x)$ is a $C^1$ function and also it is bounded (in particular, $v(0,x)$ is zero outside a bounded subset of $mathbb{R}$). Is there any theorem that allows me to state that $v(0,x)$ is integrable over $mathbb{R}$?
Thanks a lot in advance.
real-analysis compactness riemann-integration
asked Nov 20 at 7:44
eleguitar
115114
Suppose that $v=v(t,x)in C^1([0,+infty]timesmathbb{R})$ is a compactly supported function. Is it true that $$v(0,x):mathbb{R}to mathbb{R}$$ is Riemann integrable over $mathbb{R}$?
Certainly $v(0,x)$ is a $C^1$ function and also it is bounded (in particular, $v(0,x)$ is zero outside a bounded subset of $mathbb{R}$). Is there any theorem that allows me to state that $v(0,x)$ is integrable over $mathbb{R}$?
Thanks a lot in advance.
real-analysis compactness riemann-integration
real-analysis compactness riemann-integration
asked Nov 20 at 7:44
eleguitar
115114
asked Nov 20 at 7:44
eleguitar
115114
asked Nov 20 at 7:44
eleguitar
115114
asked Nov 20 at 7:44
eleguitar
115114
asked Nov 20 at 7:44
eleguitar
115114
115114
add a comment |
add a comment |
1 Answer
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If $f:mathbb R to mathbb R$ is any continuous function such that $f(x)=0$ for $|x| >N$ for some $n$ then $f$ is surely Riemann integrable on $mathbb R$ and $int_{-infty}^{infty} f(x), dx =int_{-N}^{N} f(x), dx$
answered Nov 20 at 7:47
Kavi Rama Murthy
46.9k31854
Thanks a lot. Is it also true that if $f:mathbb{R}to mathbb{R}$ is continuous and Riemann integrable on $mathbb{R}$ then $f$ is integrable over any open subset $Asubset mathbb{R}$?
– eleguitar
Nov 20 at 18:59
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
If $f:mathbb R to mathbb R$ is any continuous function such that $f(x)=0$ for $|x| >N$ for some $n$ then $f$ is surely Riemann integrable on $mathbb R$ and $int_{-infty}^{infty} f(x), dx =int_{-N}^{N} f(x), dx$
answered Nov 20 at 7:47
Kavi Rama Murthy
46.9k31854
Thanks a lot. Is it also true that if $f:mathbb{R}to mathbb{R}$ is continuous and Riemann integrable on $mathbb{R}$ then $f$ is integrable over any open subset $Asubset mathbb{R}$?
– eleguitar
Nov 20 at 18:59
add a comment |
up vote
1
down vote
accepted
If $f:mathbb R to mathbb R$ is any continuous function such that $f(x)=0$ for $|x| >N$ for some $n$ then $f$ is surely Riemann integrable on $mathbb R$ and $int_{-infty}^{infty} f(x), dx =int_{-N}^{N} f(x), dx$
answered Nov 20 at 7:47
Kavi Rama Murthy
46.9k31854
Thanks a lot. Is it also true that if $f:mathbb{R}to mathbb{R}$ is continuous and Riemann integrable on $mathbb{R}$ then $f$ is integrable over any open subset $Asubset mathbb{R}$?
– eleguitar
Nov 20 at 18:59
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
If $f:mathbb R to mathbb R$ is any continuous function such that $f(x)=0$ for $|x| >N$ for some $n$ then $f$ is surely Riemann integrable on $mathbb R$ and $int_{-infty}^{infty} f(x), dx =int_{-N}^{N} f(x), dx$
answered Nov 20 at 7:47
Kavi Rama Murthy
46.9k31854
If $f:mathbb R to mathbb R$ is any continuous function such that $f(x)=0$ for $|x| >N$ for some $n$ then $f$ is surely Riemann integrable on $mathbb R$ and $int_{-infty}^{infty} f(x), dx =int_{-N}^{N} f(x), dx$
answered Nov 20 at 7:47
Kavi Rama Murthy
46.9k31854
answered Nov 20 at 7:47
Kavi Rama Murthy
46.9k31854
answered Nov 20 at 7:47
Kavi Rama Murthy
46.9k31854
answered Nov 20 at 7:47
Kavi Rama Murthy
46.9k31854
46.9k31854
Thanks a lot. Is it also true that if $f:mathbb{R}to mathbb{R}$ is continuous and Riemann integrable on $mathbb{R}$ then $f$ is integrable over any open subset $Asubset mathbb{R}$?
– eleguitar
Nov 20 at 18:59
add a comment |
Thanks a lot. Is it also true that if $f:mathbb{R}to mathbb{R}$ is continuous and Riemann integrable on $mathbb{R}$ then $f$ is integrable over any open subset $Asubset mathbb{R}$?
– eleguitar
Nov 20 at 18:59
Thanks a lot. Is it also true that if $f:mathbb{R}to mathbb{R}$ is continuous and Riemann integrable on $mathbb{R}$ then $f$ is integrable over any open subset $Asubset mathbb{R}$?
– eleguitar
Nov 20 at 18:59
Thanks a lot. Is it also true that if $f:mathbb{R}to mathbb{R}$ is continuous and Riemann integrable on $mathbb{R}$ then $f$ is integrable over any open subset $Asubset mathbb{R}$?
– eleguitar
Nov 20 at 18:59
add a comment |
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