Real Analysis Riemann Integration

[s23fsth]

Suppose we have: f(x)= 1 if 0\leq x \leq 1 AND 2 if 1\leq x \leq 2
Using the definition, show that f is Riemann integrable on [0, 2] and find its value?

I have a general idea of how to complete this question using partitions and the L(f,P) U(f,P) definition, but am not quite receiving the answer I would like...

My workings, Fix \epsilon\succ 0 with partitions Pe = {0, 1-(\epsilon / 2), 1 + (\epsilon / 2), 2}. Then, if U(f,P)=2, we can define: L(f,P)=1(1-\epsilon/2)+1(1-\epsilon/2) = 2-\epsilon Then computing, U(f,P)-L(f,P)= 2-(2-\epsilon) =ε\precε

This doesn't seem right to me, and I am not quite sure what else to do? Any suggestions?

 

[fresh_42]

You could simply use additivity: $\int_0^2=\int_0^1+\int_1^2$ and the problem at $x=1$ will be solved.