# Math Help - f is not integrable over [0,6]

1. ## f is not integrable over [0,6]

define $f:[0,6] \to \mathbb{R}$
$f(x)=\left\{\begin{array}{ll}x&\;\;if\,\,x \ is \ rational\\0&\;\; \ x \ is \ irrational \end{array}\right.$

prove that f is not integrable over [0,6]

2. by contradiction .....

3. Originally Posted by flower3
define $f:[0,6] \to \mathbb{R}$
$f(x)=\left\{\begin{array}{ll}x&\;\;if\,\,x \ is \ rational\\0&\;\; \ x \ is \ irrational \end{array}\right.$

prove that f is not integrable over [0,6]
Exact same argument as you have seen before. It suffices to prove that isn't integrable on $[0,1]$ then given any partition $\left\{x_0,\cdots,x_n\right\}$ then $\sup_{x\in[x_{j-1},x_j]}f(x)=x_j>0,\inf_{x\in[x_{j-1},x_j]}f(x)=0$