Answer
$f$ is not integrable on $[0,1]$
Work Step by Step
Let $x_1^* = 0$ in the first subinterval of $[0,1]$
$\lim\limits_{n \to \infty}\sum_{i=1}^{n}f(x_i^*)\Delta x$
$= f(0)(\frac{1}{n})+\lim\limits_{n \to \infty}\sum_{i=2}^{n}f(x_i^*)\Delta x$
$= 0+\lim\limits_{n \to \infty}\sum_{i=2}^{n}f(x_i^*)\Delta x$
Let $~~x_1^* = c~~$ such that $\frac{1}{c} \gt n$ in the first subinterval of $[0,1]$
$\lim\limits_{n \to \infty}\sum_{i=1}^{n}f(x_i^*)\Delta x$
$= f(c)(\frac{1}{n})+\lim\limits_{n \to \infty}\sum_{i=2}^{n}f(x_i^*)\Delta x$
$= (\frac{1}{c})(\frac{1}{n})+\lim\limits_{n \to \infty}\sum_{i=2}^{n}f(x_i^*)\Delta x$
$\gt 1+\lim\limits_{n \to \infty}\sum_{i=2}^{n}f(x_i^*)\Delta x$
Since the two limits are not equal, $f$ is not integrable on $[0,1]$
