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: ∫1∗−1f(x)dx=inf{∫1−1ϕ(x)dx;ϕ:[−1,1]→Rstep function,f≤ϕ} and ∫1−1∗f(x)dx=inf{∫1−1ϕ(x)dx;ϕ:[−1,1]→Rstep function,ϕ≤f}. Now the bounded function f is Riemann integrable, if ∫1∗−1f(x)dx=∫1−1∗f(x)dx.
My first try: consider the step function ϕ:[−1,1]∖{0}→R,x↦0. It is f=ϕ everywhere on [−1,1]∖{0}. Therefore it is
∫1∗−1f(x)dx=inf{∫1−1ϕ(x)dx;ϕ:[−1,1]→Rstep function,f≤ϕ}=∫1−1ϕ(x)dx=0 and ∫1−1∗f(x)dx=∫1−1ϕ(x)dx=0. Therefore f is Riemann-integrable and ∫1−1f(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=-1
I will try next to do this with Riemann Sums. Regards
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