Hi Guys I am trying to understand what is going on as it relates to the range of the X values since from the question, the range is stated in terms of a and b. I have been exposed to a few question where the closed interval is give example([0,1] or [0,4]) but for this question I am not sure how to evaluate this function and determine if it is Riemann Integrable. Can anyone help me.
Let's review what these Darboux Sums mean.
$$\begin{align} L(f, P) &= \sum_{i=0}^{n-1} \inf_{t \in [x_i, x_{i+1}]} f(t)(x_{i+1} - x_i), \\ U(f, P) &= \sum_{i=0}^{n-1} \sup_{t \in [x_i, x_{i+1}]} f(t)(x_{i+1} - x_i). \end{align}$$ Ok. So let's pick a partition $P$ of $[0,9]$ And compute these sums.
Let $t_0=0, t_1=1/7, t_2=2/7 \dots t_{k}=k/7, t_{63}=9$. Then $P=[t_0 \dots t_{63}]$. When we are computing these Darboux Sums we look at the interval
$[5/7, 6/7]$ and we think $f(x)$ on this interval has what infimum value? The answer is $b$. Along $[5/7, 6/7]$ this function has only one value: $b$.
So we should evaluate $$\inf_{t\in [t_5,t_6]}f(t) \times 1/7=b/7$$ To make progress we have to make an assumption about $b-a$ is this value positive or negative or zero? Let's assume $a<b$. Then $L(f,P)=\frac{61}{63}b+\frac{2}{63}a$. There are these two intervals where the $\inf$ of our function evaluates to $a$. Namely and [$[t_6,t_7]=[\frac{6}{7}, 1]$ and $[t_7,t_8]=[1,\frac{8}{7}]$. Note that the $U(f,P)=b$.
So this is how to evaluate this thing on a specific partition.