Answer
$f$ is not integrable on $[0,1]$
Work Step by Step
Let $x_i^*$ be a rational number in each subinterval of $[0,1]$
$\lim\limits_{n \to \infty}\sum_{i=1}^{n}f(x_i^*)\Delta x$
$= \lim\limits_{n \to \infty}\sum_{i=1}^{n}(0)\Delta x$
$= 0$
Let $x_i^*$ be an irrational number in each subinterval of $[0,1]$
$\lim\limits_{n \to \infty}\sum_{i=1}^{n}f(x_i^*)\Delta x$
$= \lim\limits_{n \to \infty}\frac{1}{n} \sum_{i=1}^{n}(1)$
$= \lim\limits_{n \to \infty}(\frac{1}{n})(n)$
$= \lim\limits_{n \to \infty}(1)$
$= 1$
Since the two limits are not equal, $f$ is not integrable on $[0,1]$
