The following question is from GRE 8767, and I am unsure how to view this question. 
The solution says that ''In a confusing way, this question is asking us what the approximate total change in $f(x)$ is over the interval. First, the function increases by 2, then it decreases by 5, increases by 7, and decreases by 2. Adding those up gives 16.''
I struggle with the first sentence: how can I see that the supremum of this set is really just the approximate total change in $f(x)$ on the interval?
Try setting $x_0 = 0$, $x_1 = 2$, $x_2 = 5$, $x_3 = 8.5$, and $x_4 = 12$. This roughly corresponds with the locations of the local minima, local maxima, and endpoints of definition of the function. Notice that the absolute values in the summation measure the magnitude of the "swing" of the function through the given intervals. ("Up or down, I don't care; I just want to know how far you went.")
Now suppose we insert a few $x_i$s in one of these intervals. Since the function is monotonic on the interval, the sum of these "partial swings" is the full "swing" of the interval.
If we densify and densify the $x_i$s, these eventually get as close as you like to the total variation of the function, which is the sum given in the solution you quote.