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here is my question, show that the linear combination of two functions of bounded variation is also of bounded variation. Is the product of two such functions also of bounded variation?

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You don't have a better title for your question? –  J. M. Nov 30 '11 at 1:19
    
I've provided a more informative title. –  Gerry Myerson Nov 30 '11 at 1:24
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2 Answers

Hint: $f(x) g(x) - f(y) g(y) = f(x) (g(x) - g(y)) + (f(x) - f(y)) g(y)$.

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Hint: For the first one, using the definitions show that the total variation of $f + g$ is at most the total variation of $f$ plus the total variation of $g$.

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