Show that a monotone or differentiable function $h:[a,b] \to \mathbb R$ is of bounded variation on $[a, b]$.

Question

Show that a monotone or differentiable function $h:[a,b] \to \mathbb R$ is of bounded variation on $[a, b]$.

Answer 1

Suppose $h$ is monotone increasing. Let $P = (x_0,x_1,\ldots,x_n)$ be a partition of $[a,b]$.

Then

$$V_a^b(h) = \sup_{P}\sum_{k=1}^n |h(x_k) - h(x_{k-1)}|=\sup_{P}\sum_{k=1}^n \left(h(x_k) - h(x_{k-1)}\right)= h(b) - h(a).$$

If $h$ is monotone decreasing, then make a similar argument.

If $h$ is differentiable with a bounded derivative $|h'(x)| \leqslant M$, then using the MVT, there exist points $x_{k-1} \leqslant \xi_k \leqslant x_k$ such that

$$V_a^b(h) = \sup_{P}\sum_{k=1}^n |h(x_k) - h(x_{k-1)}|=\sup_{P}\sum_{k=1}^n |h'(\xi_k)|\left|x_k - x_{k-1}\right|\\ \leqslant \sup_{P}\sum_{k=1}^n M\left|x_k - x_{k-1}\right|= M[b - a].$$