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.
The function sin x oscillates between –1 and +1, and it takes those values at odd multiples of π/2. So the function G(x) oscillates between and , attaining those values at the points where is an odd multiple of π/2. However, I was wrong to say that G(x) is monotonic in the intervals between those points. Fortunately, that does not affect the argument that I gave to show that G(x) is not of bounded variation. It is still true that the variation of G(x) in each of those intervals is at least , and the sum of those numbers diverges.
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.