# How to show that a piecewise constant function is integrable, using the upper and lower sums?

Let $f(x) = \begin{cases} 1 &\mbox{if } 0\leq x<1 \\ 3 &\mbox{if } 1\leq x<2 \\ 2 &\mbox{if } 2\leq x\leq 3. \end{cases}$

Show that $f(x)$ is integrable by

• $(a)$ Finding $U(f; P_n)$ and $L(f; P_n)$ where $P_n$ is the partition of $[0; 3]$ with $\delta x = 1/n$;
• (b) Finding $\inf U(f; P_n)$ and $\sup L(f; P_n)$ and showing they're equal.

### Progress

I notice that the only difference of $U$ and $L$ are at $1$ and $2$ since $f$ is discontinuous at $1$ and $2$. How do I calculate the upper and lower sum for each interval of $[0,1)$, $[1,2)$, and $[2,3]$? As long as I know what the next step is, I think I can manage from there.

• well I notice that the only difference of U and L are at 1 and 2 since it is discontinuous at 1 and 2. How do I calculate the upper and lower sum for each interval of [0,1) [1,2) and [2,3]? As long as I know what the next step is, I think I can manage from there. Thanks – blahblah Dec 12 '14 at 5:50