I have a piecewise function and need to calculate an upper sum with $5$ subintervals.
The function in question is $$f(x)=\begin{cases}x & \text{if }0\le x < 1 \\ 2 & \text{if } x=1 \\ 4-x & \text{if }1<x\le 2\end{cases}$$
The interval is $[0,2]$, so a length of $2$. That makes $5$ subintervals, each with a length of $2/5$. The subintervals are $[0,2/5],[2/5,4/5],[4/5,6/5],[6/5,8/5]$ and $[8/5,2]$. I have managed to find suprema and infima for all other intervals except $[4/5,6/5]$ since the function is discontinuous at $1$.
I'm pretty sure that the infimum would be $f(4/5)=4/5$, since $f$ is defined at this point. Defining the supremum is a bigger issue though. It can't be $f(1)=2$, since the function obtains values larger than this on the subinterval, but the same issue arises with $f(6/5)$... Any help is appreciated!