Question

\(f(x) =\begin{arrays}x~~if ~~x\in \mathbb Q\\1~~if~~x\notin\mathbb Q\end{arrays}\)
a) Let P be a partitin of [0,1]. Find \(lim_{||P||\rightarrow 0}U(f,P)\) and \(lim_{||P||\rightarrow 0}L(f,P)\) and conclude that f is not Riemann integrable on [0,1]
b) Repeat on [0,2] with partition P of [0,2] that contains the point 1 and draw a conclusion regarding Riemann integrability of f on[0,2]
Please, help @wio

Answer 1

I got a)

Answer 2

Divide [0,1] into n parts with length 1/n , hence as \(n\rightarrow \infty\), \(||P||\rightarrow 0\)

Answer 3

0.0 i dont know math

Answer 4

on[0,1], if x in Q, x<1, hence upper sum is \(U(f,P)=\sum_{i=1}^n 1\triangle x=1\)
and lower sum is \(L(f,P) =\sum_{i=1}^n x\triangle x = x\)
hence limit of them are different , hence f is not Riemann integrable on [0,1]

Answer 5

But I don't know how to do b) because on [1,2] if x in Q , f(x) =x and it is not < 1

Answer 6

@wio @dan815 @myininaya