Monday, 21 April 2014

real analysis - Some questions about Riemann integration

I just learn some basic definition about



Let f,g:[0,1]R be




f(x)={1,if xQ0,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 xD0,otherwise



Can we define a similar 'integral' to say the value of the 'integral' = 1/2



Thank you!

No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...