Trying to imagine how functions of bounded variation look like I am trying to wrap my head around what functions of bounded variation mean or how they exactly look like. On Wikipedia I read that

For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value.

But if I define my function $\sin(x)$ on the interval $(-\infty, \infty)$ the value also will be infinite (while I move the point along the graph the sum of the distances along the y axis will go to infinity)?
 A: The main crude intuition behind BV one dimensional maps is that on a compact interval you want to have “finite variation”, that roughly means the map “cannot oscillate too much”: you cannot be able to find a sequence of finite partitions on the x-axis such that the image values of the sequence of partitions go to infinity.
Note that a function of bounded variation (BV henceforth) in one dimension may not be continuous and a continuous function may not be BV on a compact interval. For the counterexamples just take any monotone function on a closed interval and the following function defined on $[0,1]$ $$ u(x)=x^a\sin(\frac{1}{x^b}) $$ defined to be zero at zero
is not of bounded variation as long as $1 \le a\leq b$.
Nevertheless, BV functions have nice differentiability properties, at least if you are willing to accept throwing away a “measure theoretic” small set of points. Indeed, a function of bounded variation on a compact interval is necessary differentiable a.e., so that any continuous function which is not differentiable on a positive measure set is not of bounded variation.
Also, some function which oscillates too much can not be bounded variation as the example above shows.
So, the notions of continuity and BV are kind of transversal: the best way to think about it is in terms of “controlled oscillations” on a compact set.
