I'm self-studying the theory of functions of bounded variations from the book of Neal Carother (Real Analysis) and need some help with this problem:
Show that $V_{a}^{b}(X_\mathbb{Q}) = +\infty$ on any interval [a, b].
In the previous chapter the author defines $X_\mathbb{Q}$, the characteristic function of $\mathbb{Q}$, by $X_\mathbb{Q}(x) = 1$ if $x \in \mathbb{Q}$ and $X_\mathbb{Q}(x) = 0$ if $x \notin \mathbb{Q}$.
My thoughts:
Let $P_n$ be a random partition of $(n + 1)$ points of the closed interval $[a, b]$. That is $$P_n = [a, x_1, x_2, ..., x_{n-1}, b].$$ The total variation of the characteristic equation of $\mathbb{Q}$ over the interval $[a, b]$ can be written as $$V_{a}^{b}(X_\mathbb{Q}) = V_{a}^{x_1}(X_\mathbb{Q}) + V_{x_1}^{x_2}(X_\mathbb{Q}) + ... + V_{x_{n - 1}}^{b}(X_\mathbb{Q})$$ but I don't know how to prove that this sum tends to infinity.