# Trying to imagine how functions of bounded variation look like

analysisbounded-variationtotal-variation

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)?

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 $$1 \le a\leq b$$.