I guys I have a general question how can I see if a function is of bounded variation.
For example $h(x)=x^2(sin(\frac{1}{x})$ for all x except if $x=0$ then $h(0)=0$
SO I know this is of bounded variation on the interval of $[-1,1]$ because it was noted in class. But I do not see why and how do I see if a function is of BV in general. Is there a general idea of what to look at?
def of function of bounded variation is $T_a ^b < \infty$ when $T_a ^b =sup\{ \sum _{i=1}^n|f(x_i)-f(x_{i-1})|\}$
In some cases, you'll be able to pick out where the local maximums and minimums are exactly. Choose these for your $x_i$, and note that no other choice would have greater variation. Then it's basically a calculus problem to show that the sum of these lengths converges.
I suspect this is the case for the example you did in class -- $\sin(1/x)$ has easy to find maximums and minimums. Then simply multiply by $x^2$ at the end.
Otherwise, you'll have to simply try to bound it by something that converges. For example, if you say the lengths you're summing $|f(x_i)-f(x_{i-1})|$ are each no more than some quantity, and the sum still converges, you're still fine.