Is there an error in my textbook? 
The weird thing is, the last sentence says, for the case where $x_1$, "since $\Delta x_1$ approaches zero as $\max \Delta x_k \rightarrow 0...$". But given that $\Delta x_1$ is formed between $x_0$ and $x_1$, and if $x_1$ is equal to $0$ and the question says $[0,1]$, wouldn't $\Delta x_1 = 0$ at all times?

If the partition is $\{0,\ 1/6,\ 3/6,\ 4/6,\ 5/6,\ 1\}$ then $\Delta x_1= \dfrac 1 6 - 0 = \dfrac 1 6 \ne 0$.
In your first case, $x_1^* \ne 0$ and in your second case $x_1^*=0$.  But in either case $\Delta x_1 = x_1 - x_0$ need not be $0$.  However, if it is $0$, then the statement that it approaches $0$ would still be true.
