I have a little confusion about this example of bounded variation :

Consider the function $f(x)=\frac{1}{x}$ in the interval $(0,1)$.

As the function $f$ is monotone decreasing in $(0,1)$ so it is Bounded Variation.

Again the function is NOT bounded in $(0,1)$. So it is NOT of Bounded Variation.

So what I conclude about $f$ ?