I have the following question:
Let $f$ be a function defined in the following interval: [0,1], such that:$$f(x)=\begin{cases} 2 & 0\le x<\frac{1}{3} \\\\ 0 & \frac{1}{3}\le x<\frac{2}{3} \\\\ 1 & \frac{2}{3}\le x<1 \end{cases}$$ Is the given function Riemann integrable? if so calculate the integral via Riemann integral's definition.
My attempt:
My intuition says she is Riemann integrable, and the value of the integral is-1 since we can calculate the rectangles in the graph. However, I find it hard to prove that via Riemann integral's definition, because I cannot choose an equal partition $P$ for the upper and lower Darboux's sums to show $\sup_P L(P,f) = \inf_P U(P,f)$.
If $\varepsilon>0$, consider the partition $P_\varepsilon=\left\{0,\frac13-\frac\varepsilon4,\frac13,\frac23-\frac\varepsilon2,\frac23,1\right\}$. Then$$U(P_\varepsilon,f)=1\quad\text{and}\quad L(P_\varepsilon,f)=1-\varepsilon.$$Therefore, $f$ is Riemann-integrable, and $\int_0^1f=1$.