Let $f: [0,2] \to \mathbb{R}$ be defined by $$ f(x) = \begin{cases} x^2 & \text{if } 0 \le x < 1, \\ 0 & \text{if } 1 \le x \le 2. \end{cases} $$ For any $\epsilon>0$, let $P_\epsilon = \{ 0, 1-\epsilon, 1, 2 \}$ be the partition of $[0,2]$. Compute $U(f,P_\epsilon) - L(f,P_\epsilon)$. Is $f$ Riemann integrable? If yes, compute $\int_0^2 f$.
This is what I did. $$U(f, P_\epsilon) = (1-\epsilon) \sup_{[0,1-\epsilon]} f + \epsilon \sup_{[1-\epsilon,1]} f + 1 \cdot \sup_{[1,2]} f = (1 - \epsilon)^3 + \epsilon,$$ $$L(f, P_\epsilon) = (1-\epsilon) \inf_{[0,1-\epsilon]} f + \epsilon \inf_{[1-\epsilon,1]} f + 1 \cdot \inf_{[1,2]} f = 0.$$ Is it right? I think my work was wrong. For me, this function is Riemann integrable, and I had to choose a different partition (from what the problem gave me) to make $U(f,P_\epsilon)-L(f,P_\epsilon) < \epsilon$ (and go to $0$ as $\epsilon \to 0$). What should $P_\epsilon$ be?
Thank you.