What is the background for $\sum_{k=1}^n|f(x_k)-f(x_{k-1})|$? The question is from the following problem:



If $f$ is the function whose graph is indicated in the figure above, then the least upper bound (supremum) of
$$\big\{\sum_{k=1}^n|f(x_k)-f(x_{k-1})|:0=x_0&lt;x_1&lt;\cdots&lt;x_{n-1}&lt;x_n=12\big\}$$
appears to be  

$A. 2\quad B. 7\quad C. 12\quad D. 16\quad E. 21$

I don't know what the set above means. And I am curious about the background of the set in real analysis.
If the supremum is finite, the function is called of bounded variation. 

Function of bounded variation appear in some context. A simple example I like is the following: a function is of bounded variation if and only if it is  the sum of an increasing and a decreasing function.
Total variation sums up how much a function bobs up and down. Yours does this 16 units.  Therefore choose D.