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







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    0














    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







    share|cite|improve this question

























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      0







      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







      share|cite|improve this question













      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






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      asked Nov 20 at 7:44









      eleguitar

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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 $int_{-infty}^{infty} f(x), dx =int_{-N}^{N} f(x), dx$






          share|cite|improve this answer





















          • 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











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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 $int_{-infty}^{infty} f(x), dx =int_{-N}^{N} f(x), dx$






          share|cite|improve this answer





















          • 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
















          1














          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 $int_{-infty}^{infty} f(x), dx =int_{-N}^{N} f(x), dx$






          share|cite|improve this answer





















          • 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














          1












          1








          1






          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 $int_{-infty}^{infty} f(x), dx =int_{-N}^{N} f(x), dx$






          share|cite|improve this answer












          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 $int_{-infty}^{infty} f(x), dx =int_{-N}^{N} f(x), dx$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 20 at 7:47









          Kavi Rama Murthy

          49.6k31854




          49.6k31854












          • 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




          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


















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