Let and with domain (when both functions are said to be equal to 0).
I can show that both and are differentiable at every point. However, I don't know how to show that is of bounded variation while is not. I am only allowed to use the definition in my proof.
Any hints as where to start would be greatly appreciated.
For F(x), the situation is more serious, and the argument that I gave before does not work without some modification. What you can say is that the turning points of F(x) occur at the points where (easy calculus calculation). There is exactly one such point in each of the intervals . So F has only one turning point in each such interval. The variation of F in that interval is therefore at most twice the maximum difference between and in the interval, namely . That's sufficient to show that F is of bounded variation.