Trying to imagine how functions of bounded variation look like

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.