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Let $f$ be a function of bounded variation on $[a, b]$ and $T_{a}^{b}(f)$ its total variation. We do not assume that $f$ is continuous. Show that \int_{a}^{b}|f'(t)|\, dt \leq T_{a}^{b}(f).

I know that if we assume that $f$ is continuous, then the above equation is true because we have the ability to use the Mean Value Theorem. What can I do if we don't assume $f$ is continuous?

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    But since $f$ is of bounded variation, $f'$ exists almost everywhere and so $f$ is continuous almost everywhere. But I need $f$ to be continuous everywhere to use the MVT.2012-03-31

1 Answers 1

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Since $f$ is of bounded variation, we can write $f = g -h$ where $g$ and $h$ are monotone increasing and f' = g' -h' a.e. and |f'| \leq |g'| + |h'| = g' + h'. So

\int_a^b |f'| \leqslant\int_a^b g' + \int_a^b h' \leqslant \ldots

Can you take it from here?

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    @938049: the first part of what you wrote is true for increasing real-valued functions on $[a,b]$ that are differentiable a.e. So it works.2012-03-31