- #1

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## Homework Statement

Given a sequence of scalars (c

_{n}) and a sequence of distinct points (x

_{n}) in (a, b), define f(x) = c

_{n}if x = x

_{n}for some n, and f(x) = 0 otherwise. Under what condition(s) is f of bounded variation on [a,b]?

## Homework Equations

V

^{b}

_{a}f = sup

_{p}([tex]\Sigma[/tex]lf(t

_{i}) - f(t

_{i-1})l< +inf, then f is of Bounded Variation.

## The Attempt at a Solution

My understanding of the question is that we have points at x in a vertical line and points along y=0 at every other value of x. So the only way that this function is of bounded variation is if there is a supremum of the constants so that they can't go on vertically to infinitum. If we merely say that they are bounded above we won't have bounded variation, so we must say that the supremum is a member of the set of c

_{n}'s.