Bounded variation

Remarks

Example: a nondecreasing function $f$ on $[a, b]$ has bounded variation. For every partition the sum telescopes: $V(f, P) = \sum_i (f(x_i) - f(x_{i-1})) = f(b) - f(a)$, so $V_a^b(f) = f(b) - f(a)$.

Non-example: let $f(0) = 0$ and $f(x) = x \sin(1/x)$ for $0 < x \le 1$. The function $f$ is continuous, but it does not have bounded variation. At the points $x_k = 1/((k + \frac12)\pi)$ the values are $\pm x_k$ with alternating signs, so a partition through $x_1, \dots, x_N$ gives $V(f, P) \ge \sum_{k=1}^{N} x_k$, and this sum diverges like the harmonic series.

The set of functions of bounded variation on $[a, b]$ is often written $BV([a, b])$. Every function of bounded variation is bounded, since $\abs{f(x)} \le \abs{f(a)} + V_a^b(f)$.