Question

Find the upper and lower Riemann integrals of the function f , defined on [a,b]

as follows:


0 when is x irrational

f(x)={

2 when is x irrational


Is f Riemann integrable on [ a,b]?Justify your answer.

Answer 1

Let's assume that $f=2$ if $x\in Q$

For any partition $0<x_1<x_2<...x_n<1$ the Riemann's upper integral is

$\lim\limits_{n\to\infty}\sum\limits_{k=1}^n(x_k-x_{k-1})\sup\limits_{x_{k-1}<x<x_k}f(x)=2\lim\limits_{n\to\infty}\sum\limits_{k=1}^n(x_k-x_{k-1})=2$

because in each interval $(x_{k-1};x_k)$ there is a rational number $q$ and hence the supremum is 2.

The lower integral is

$\lim\limits_{n\to\infty}\sum\limits_{k=1}^n(x_k-x_{k-1})\inf\limits_{x_{k-1}<x<x_k}f(x)=0$

because in each interval $(x_{k-1};x_k)$ there is an irrational number $\alpha$ and hence the infimum is 0.

Since upper and lower interals are not equal, the function is not Riemann integrable.