Riemann Integrability - Piecewise function

Question

For $f(x) = \begin{cases} 1, & 0\le x\le1 \\ 2, & 1<x\le2 \end{cases}$

Use the definition of the Riemann Integral to show that $f$ is Riemann Integrable over $[0,2]$.

It was suggested that I use the partition $(0, 1-\epsilon, 1+ \epsilon, 2)$ and let $ \epsilon\rightarrow 0$. Unfortunately, I am unsure how to do this. Any guidance would help.

Answer 1

The idea using the partition of the hint, $P=(t_{0}=0,t_{1}=1-\epsilon,t_{2}=1+\epsilon,t_{3}=2)$, is that$\\$ U(f,P)-L(f,P)=$\sum_{0}^{2}(M_{s}(f)-m_{s}(f))(t_{k+1}-t_{k})=2\epsilon$. Therefore if $\epsilon$ tends to $0$ we have U(f,P)=L(f,P) then f is integrable.