Let f(x)={0 x rational ; 1 x irrational} Let P be any partition of [0,1]. Compute L(f,P) and U(f,P)/ Compute L(f) and U(f). Is f integrable on [0,1]
consider any interval $I$ of width $\delta$ that's a subset of $[0,1]$
does $I$ contain a rational number?
If so what will $L(f,I)$ be?
does $I$ contain an irrational number?
If so what will $U(f,I)$ be?
now consider a partition of $[0,1]$ which will be made up of intervals such as $I$ above
given what you know about $L,U$ of $I$ what is $L(f,P),~U(f,P)$ ?
If $f$ is integrable on $[0,1]$ what is required of $L(f,P)$ and $U(f,P)$ ?
Does that hold true in this case?
So is $f$ integrable on $[0,1]$ ?