It was familiar to me to deal with continuous functions in proving Riemann integrability, so I wanna show my work in solving this question (maybe in a messy way) and I'm waiting for any advice or hint.
I chose $\delta_{x_i}= \frac{1}{n}$.
We have $m_i= 1$, $\forall$ $i$.
$ M_i = \begin{cases} 1 & \quad \text{if } \text{ $x_i$ doesn't belong to a subinterval containing 1}\\ 2 & \quad \text{if } \text{ $x_i$ belongs to a subinterval containing 1}\\ \end{cases}$
So $L(f,P)= \frac{1}{n} \sum_{i=1}^n 1$ $=1$.
$U(f,P)= \frac{1}{n}.2 + \frac{1}{n}.2 + \sum_{i=1}^{n-2} 1$ $= \frac{n-1}{n} + \frac{4}{n}$.
So we have $|U(f,P)- L(f,P)|$ $= \frac{n-1}{n} + \frac{4}{n} - 1$ $\longrightarrow 0$. Therefore, the function is Riemann integrable.
I feel that I'm catching the idea but not dealing well with it. Please guide me the right track and the right way of writing.
Thanks!