Sunday, 27 July 2014

integration - function which is Riemann integrable

Consider f:[1,1]R, x{1,if x=00else  I want to know why f is Riemann integrable and I tried something. First of all we had the definition in lecture: 11f(x)dx=inf{11ϕ(x)dx;ϕ:[1,1]Rstep function,fϕ} and 11f(x)dx=inf{11ϕ(x)dx;ϕ:[1,1]Rstep function,ϕf}. Now the bounded function f is Riemann integrable, if 11f(x)dx=11f(x)dx.



My first try: consider the step function ϕ:[1,1]{0}R,x0. It is f=ϕ everywhere on [1,1]{0}. Therefore it is
11f(x)dx=inf{11ϕ(x)dx;ϕ:[1,1]Rstep function,fϕ}=11ϕ(x)dx=0 and 11f(x)dx=11ϕ(x)dx=0. Therefore f is Riemann-integrable and 11f(x)dx=0. Could you help me, to correct my solution?



I'm not sure if it is ok, because my step function isn't defined in x=0. But we say that ϕ:[1,1]R is a step function if you have a partition of the interval [1,1], $x_0=-1Maybe I have to define ϕ in x=0 too but take x=0 as one of the xis.



I will try next to do this with Riemann Sums. Regards

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