1)How does the graph of a funtion of bounded variation behave. 2)Why a bounded function is not always a function of bounded variation.Please explain graphically. 3)What purpose does bounded variation serve.I mean why they are defined. Sorry if the question is already asked. And thank you in advance.
2026-03-27 19:52:34.1774641154
Please explain to me the following.
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1
On a interval, the length of the function's curve is finite.
Think of $\sin x^{-1}$ on $]0,1]$.
Let $f$ be of bounded variation. It has the following properties:
It is the difference of two bounded increasing functions.
The left and right limits exists at every point within $f$'s domain.
The discontinuity set of $f$ is countable, and $f$ is thus Riemann integrable.
$f'$ is Lebesgue integrable.