I am reading about Riemann-Stieltjes Integration in Carother's Real Analysis. We find that functions of Bounded Variation provide us with a rich class of integrators. Therefore, I am trying to learn how to quickly identify whether a function is of bounded variation. So far I have been using Jordan's Theorem is quite helpful.
Jordans's Theorem
If $f = \alpha – \beta$ where $\alpha,\beta$ are non-decreasing functions, then $f \in BV([a,b])$.
I used Jordan's Theorem to solve the following question below:
Question 1) Determine whether or not $f(x)=x^{2}sin(\frac{1}{x})$ if $x \neq 0$ and $f(0)=0$ is of bounded variation on $[0,1]$.
Solution 1): By Jordan's Theorem, we aim to find two non-decreasing functions, $\alpha,\beta$ such that $f(x) = x^{2}sin(\frac{1}{x})$ $=$ $\alpha – \beta$. Intuitively, I think to myself that $f(x)$ is bounded above by 1, therefore when trying to find a non-decreasing $\alpha$, I could add a non-decreasing function, say $10x$, that "dominates" $f(x)$ so that $f(x) + 10x$ is non-decreasing also. In other words, because $10x$ "dominates" $f(x)$ and $10x$ is non-decreasing, then their sums will be non-decreasing. We can then simply make $\beta = 10x$, which is non-decreasing itself to get $f(x) = \alpha – \beta$ which is the difference of two non-decreasing functions. We are done.
Now here is my problem.
Question 2) Determine whether or not $f(x) = \sqrt{x}sin(\frac{1}{x})$ if $x \neq 0$ and $f(0)=0$ is of bounded variation on $[0,1]$.
Solution 2): I would reuse the technique in Solution 1).
The problem is $f(x)$ in Question 2 is not of bounded variation. Therefore, my reasoning above must be incorrect somewhere but I cannot seem to find it. Also, let's say I knew that this function is not of bounded variation. Then I would aim to show the following:
\begin{align*}
V_{0}^{1}(f) = \underset{P}{Sup}\sum_{k=1}^{n}|f(x_{i})-f(x_{i-1})| \overset{n\to\infty}{\longrightarrow}\infty
\end{align*}
However, I can't seem to find a partition P that satisfies the statement above. I'm thinking that there is some special sequence of $x's$ that would make the summation above equal some sequence of values we know blows up to $\infty$. Any clarification would be greatly appreciated.
Best Answer
Hint: Take $x_{2k}=\frac{1}{\pi k}$ and $x_{2k+1}=\frac{1}{2\pi k+\frac{\pi}{2}}$