Okay, so I have a hard time understanding this math problem:
Given
$$[f(x)] = \begin{cases} 5 & \quad \text{if } x \text{ <3}\\ 7 & \quad \text{if } x \geq3 \end{cases} $$ were the partitioning is $$P_n=[0,3-\frac{1}{n},3+\frac{1}{n},4]$$ where $$n\in N$$ and $$I=[0,4]$$ make the graph to f and calculate $$L(f P_n) , U(f,P_n)$$ But I don't understand how the formula for $$P_n$$ works. Could someone give me an example of how to use it?
EDIT: How do I find out if f(x) is integrable or not? Is this right:
Since epsilon must be bigger than $$0$$ (or I assume so), and $$U(f,P_n)-L(f,P_n)=-\frac{4}{n}<\epsilon$$ the function must be integrable on [0,4]
Let $x_0=0,\,x_1=3-\frac{1}{n},\,x_2=3+\frac{1}{n},\,x_3=4$. Then by definition $$L(P_n,f)=\sum_{m=1}^3\inf_{x\in[x_{m-1},x_m]} f(x)(x_m-x_{m-1}),\text{ and } U(P_n,f)=\sum_{m=1}^3\sup_{x\in[x_{m-1},x_m]} f(x)(x_m-x_{m-1})$$. Hence computing the supremum and infimum accordingly, we get $$L(P_n,f)=5\Big(3-\frac{1}{n}\Big)+5\Big(\frac{2}{n}\Big)+7\Big(1-\frac{1}{n}\Big)=22-\frac{2}{n},$$ $$U(P_n,f)=5\Big(3-\frac{1}{n}\Big)+7\Big(\frac{2}{n}\Big)+7\Big(1-\frac{1}{n}\Big)=22+\frac{2}{n}.$$