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? 

Answer 1

Let $f(x)$ and $g(x)$ be two functions of bounded variations.

So, the sum i.e., $f(x)+g(x)$

We can get $h(x)=f(x)+g(x)$

Using the definition of a function of continuous variation we know that the total variation is atmost the variation of $f(x)+g(x)$

So, $h(x)=f(x)+g(x)$ is a function of continuous variation.

Now, let $c(x)=f(x)\cdot g(x)$

We can re-write the function as:-

$|(fg)(x)-(fg)(y)|\leq |f(x)||g(x)-g(y)|+|g(y)||f(x)-f(y)|\>\> [triangle\>inequality]$

So, this is a transposition of the previous function i.e., the sum of the functions.

So, we conclude that the product of two functions is also of bounded variation.