Answer to Question #170285 in Real Analysis for Prathibha Rose

Question

Let alpha be of bounded variation on [a,b] and assume that f element of R(alpha) on [a,b] .show that f element of R(alpha) on every subinterval [c,d] of [a,b]

Answer 1

We can see this by a counterexample. Consider the following function,

$f(x)=\dfrac{1}{1-x}when$ $x\neq1;$

and $f(x)=1$ when $x=1.$

This function is increasing on (0, 1) and so on every closed subinterval of (0, 1) it is of bounded variation by Theorem "If f is increasing on [a, b], then f is of bounded variation on [a, b] and

V (f, [a, b]) = f(b) − f(a)".

However, because it has a vertical asymptote at x = 1 we can make the sum $\Sigma_{i=1}^{n}|f(x_i)-f(x_{i-1})|$ as large as we like by choosing partition points close to 1. Thus V (f, [0, 1]) = ∞ and f is not of bounded variation on [0, 1].

The following example is especially interesting because it shows that a continuous function need not be of bounded variation.