# Is the function riemann integrable?

Define

$$f(x)= \begin{cases} 3 & 0 \le x \le 2 \\ 2 & 2

over $$[0,6].$$ Is this function Riemann integrable over $$[0,6]$$?

I believe that it is, but I'm not sure if my solution is correct.

Basically, I considered the partition of the interval: $$\{0, 2 - \delta, 2 +\delta, 3 -\delta, 3 + \delta, 6 \}$$ and showed that over this partition, the lower sum is $$20 - 3 \delta$$ and the upper sum is $$20 + 3 \delta$$.

Since the difference is only $$6 \delta$$, we can make delta arbitrarily small, meaning the function is integrable.

Is this correct?

• Your method reminds of the proof of a more general result. You can have the same conclusion for any function that is bounded on $[0,6]$ having only a finite number of discontinuities within $(0,6)$. Dec 6, 2021 at 2:36
• another route: it is monotonic on $[0,3]$, it is monotonic on $[3,6]$ Dec 6, 2021 at 2:45
• Riemann integrable if a function is piecewise continuous. Dec 6, 2021 at 2:45

Yes, your answer is correct by the Cauchy criterion of Riemann Integrability. Your function $$f$$ is bounded over $$[0,6]$$, and you can take $$\delta = \dfrac{\epsilon}{6}> 0$$. The partition you used works !