Question

If f is monotonic on [a,b], then show that f is of bounded variation on [a,b]. 

Answer 1

A function$f : [a,b] →R$ is monotonic if and only if $f$ is a function with bounded variation and $\overset{b}{ \underset{a}{V}} = | f(b) – f(a) |.$

A function$f : [a,b] →R$ is called with bounded variation on $[a,b]$ if there is $M > 0$ such that for any partition $∆ = (a = x_0 < x_1 < ... < x_n = b)$ of the interval [a,b] we have:

$\overset{b}{ \underset{a}{V}} = sup \{V_∆(f) |∆ \ division \ of\ [a,b]\}$

is called the total variation of the function f on the interval $[a,b]$ .

Given a monotonic function $f: [a,b]→R$ and a partition $P=\{a=x_0< x_1< x_2< ... < x_n< b=x_n+1\}$

of $[a,b]$, the variation of $f$ over $P$ is;

$$V_P(f) =∑_{j=0}^n|f(x_j)−f(x_{j+1})|.$$

$f$ is of bounded variation if the numbers $V_P(f)$ form a bounded set, as $P$ ranges over the set of all partitions of $[a,b]$. We denote the supremum of the $V_P(f)$ over all partitions $P$ by $V_{ab}(f)$, the variation of $f$ from $a$ to $b$.