Question:Suppose $g$ is continuous on $[a,b]$. Let f(x)=$\int_{a}^{x}g$ where $x∊[a,b]$. Show that $\int_{a}^{x}|g|$ gives the total variation of $f$ on $[a,x]$.
I managed to prove that $V_{f}(a,x)≤\int_{a}^{x}|g|$. But I still could not find a way to prove that $V_{f}(a,x)≥\int_{a}^{x}|g|$. I would much appreciate if someone could provide me a hint. Thanks
Hint: The integral $\int_a^x |g|dx$ is defined to be the supremum of all Riemann sums of $|g|$ over $[a,x]$. Given a Riemann sum, find a partition $P$ for which the variation of $|g|$ over $P$ is greater than this sum. From there, we may conclude that the supremum of Riemann sums is less than the supremum of variations over partitions, yielding the necessary conclusion.
