How to solve Stieltjes integral $int_0^n f(x) d lfloor x rfloor$?

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I'm trying to solve the following



$$
int_0^n f(x) d lfloor x rfloor
$$



where $lfloor x rfloor$ is the floor function, $f : [0,n] rightarrow mathbbR$ is continuous, and $n in mathbbN$.



I know $fracddx lfloor x rfloor$ itself cannot be secured (i.e. not differentiable). I cannot proceed further.



Can anyone give some hints?






calculus integration stieltjes-integral







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  • The answer is $f(1)+...+f(n-1)$. Just write down Riemann - Steiltjes sums and take the limit.
    – Kavi Rama Murthy
    Aug 31 at 8:06







  • 2




    @KaviRamaMurthy Shouldn't it be $$int_0^n,f(x),textdlfloor xrfloor = f(1)+f(2)+ldots+f(n),?$$ I think we have $$int_0^n,f(x),textdlceil xrceil=f(0)+f(1)+ldots+f(n-1),.$$
    – Batominovski
    Aug 31 at 8:08















up vote
0
down vote

favorite












I'm trying to solve the following



$$
int_0^n f(x) d lfloor x rfloor
$$



where $lfloor x rfloor$ is the floor function, $f : [0,n] rightarrow mathbbR$ is continuous, and $n in mathbbN$.



I know $fracddx lfloor x rfloor$ itself cannot be secured (i.e. not differentiable). I cannot proceed further.



Can anyone give some hints?






calculus integration stieltjes-integral







share|cite|improve this question





















  • The answer is $f(1)+...+f(n-1)$. Just write down Riemann - Steiltjes sums and take the limit.
    – Kavi Rama Murthy
    Aug 31 at 8:06







  • 2




    @KaviRamaMurthy Shouldn't it be $$int_0^n,f(x),textdlfloor xrfloor = f(1)+f(2)+ldots+f(n),?$$ I think we have $$int_0^n,f(x),textdlceil xrceil=f(0)+f(1)+ldots+f(n-1),.$$
    – Batominovski
    Aug 31 at 8:08













up vote
0
down vote

favorite









up vote
0
down vote

favorite











I'm trying to solve the following



$$
int_0^n f(x) d lfloor x rfloor
$$



where $lfloor x rfloor$ is the floor function, $f : [0,n] rightarrow mathbbR$ is continuous, and $n in mathbbN$.



I know $fracddx lfloor x rfloor$ itself cannot be secured (i.e. not differentiable). I cannot proceed further.



Can anyone give some hints?






calculus integration stieltjes-integral







share|cite|improve this question













I'm trying to solve the following



$$
int_0^n f(x) d lfloor x rfloor
$$



where $lfloor x rfloor$ is the floor function, $f : [0,n] rightarrow mathbbR$ is continuous, and $n in mathbbN$.



I know $fracddx lfloor x rfloor$ itself cannot be secured (i.e. not differentiable). I cannot proceed further.



Can anyone give some hints?






calculus integration stieltjes-integral




calculus integration stieltjes-integral






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 31 at 8:01









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  • The answer is $f(1)+...+f(n-1)$. Just write down Riemann - Steiltjes sums and take the limit.
    – Kavi Rama Murthy
    Aug 31 at 8:06







  • 2




    @KaviRamaMurthy Shouldn't it be $$int_0^n,f(x),textdlfloor xrfloor = f(1)+f(2)+ldots+f(n),?$$ I think we have $$int_0^n,f(x),textdlceil xrceil=f(0)+f(1)+ldots+f(n-1),.$$
    – Batominovski
    Aug 31 at 8:08

















  • The answer is $f(1)+...+f(n-1)$. Just write down Riemann - Steiltjes sums and take the limit.
    – Kavi Rama Murthy
    Aug 31 at 8:06







  • 2




    @KaviRamaMurthy Shouldn't it be $$int_0^n,f(x),textdlfloor xrfloor = f(1)+f(2)+ldots+f(n),?$$ I think we have $$int_0^n,f(x),textdlceil xrceil=f(0)+f(1)+ldots+f(n-1),.$$
    – Batominovski
    Aug 31 at 8:08
















The answer is $f(1)+...+f(n-1)$. Just write down Riemann - Steiltjes sums and take the limit.
– Kavi Rama Murthy
Aug 31 at 8:06





The answer is $f(1)+...+f(n-1)$. Just write down Riemann - Steiltjes sums and take the limit.
– Kavi Rama Murthy
Aug 31 at 8:06





2




2




@KaviRamaMurthy Shouldn't it be $$int_0^n,f(x),textdlfloor xrfloor = f(1)+f(2)+ldots+f(n),?$$ I think we have $$int_0^n,f(x),textdlceil xrceil=f(0)+f(1)+ldots+f(n-1),.$$
– Batominovski
Aug 31 at 8:08





@KaviRamaMurthy Shouldn't it be $$int_0^n,f(x),textdlfloor xrfloor = f(1)+f(2)+ldots+f(n),?$$ I think we have $$int_0^n,f(x),textdlceil xrceil=f(0)+f(1)+ldots+f(n-1),.$$
– Batominovski
Aug 31 at 8:08
















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