How to check a function whether it is of bounded variation or not in an open interval.

Question

How to check a function whether it is of bounded variation or not in an open interval.

$f(x) = \sqrt{(1 - x^2)}$ Is this function of bounded variation on $x \in (-1,1)$?

My thought: I know how to check it in $[-1,1]$. This function will be monotone on both the interval $[-1,0]$ and $[0,1]$.So this will be of bounded variation on $[-1,1]$.

Can anyone please help me?

Answer 1

$f'(x)=-2x\frac 1 {2\sqrt {1-x^{2}}}$ is integrable on $(-1,1)$ so the total variation of $f$ on $(-1,1)$ does not exceed $\int_{-1}^{1} |f'(x)|\, dx$.