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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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    There are some comments [here](http://math.stackexchange.com/q/68364/) that may help.2012-03-31
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    If $f^\prime$ exists, then $f$ is continuous.2012-03-31
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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

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