I just learn some basic definition about
Let f,g:[0,1]→R be
f(x)={1,if x∈Q0,otherwise
g(x)={1,if x=1/n,n=1,2,...0,otherwise
We know f is not Riemann integrable, but g is.
So my first question is, is it true that if the set of discontinuous points is a dense set, then that function is not Riemann integrable.
My second question is we know h:[0,1]→R by h(x)=1 is integrable and has value 1. So if we have a dense set D in [0,1] which cardinality of D and Dc are equal, and define u(x)={1,if x∈D0,otherwise
Can we define a similar 'integral' to say the value of the 'integral' = 1/2
Thank you!
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