Prove that a product of functions of bounded variation is a function of bounded variation

calculus calculus-of-variations

We consider functions defined on an interval $[a,b]$. I have to prove that a product of functions of bounded variation is a function of bounded variation. I have to also show that this isn't true for quotient in general and tell which additional assumption guarantees that quotient IS of bounded variation.


Solution 1:

Hint:

$|(fg)(x)-(fg)(y)|\leq |f(x)||g(x)-g(y)|+|g(y)||f(x)-f(y)|$. Again $f,g$ are bounded so what will you get from here??

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