# Functions of finite square variation

1. Jan 24, 2005

### leezly

Suppose we have two continuous functions f having finite square variation and g having zero square variation on compacts, i.e.:
- let U(n)={0=t(0),t(1),...,t(n-1)} be a partition of the line [0,t] with t(n)=t, t fixed and
Code (Text):
$$\lim_{n\to\infty}\max_{t(i)\in U(n)}|t(i+1)-t(i)|=0$$
- then
Code (Text):
$$\lim_{n\to\infty}\sum_{t(i)\in U(n)}\Big(f\big(t(i+1)\big)-f\big(t(i)\big)\Big)^2=V;\ V\in\mathbb{R}\quad\mbox{and}\quad\lim_{n\to\infty}\sum_{t(i)\in U(n)}\Big(g\big(t(i+1)\big)-g\big(t(i)\big)\Big)^2=0$$
The question:
Do we then know, that
Code (Text):
$$\lim_{n\to\infty}\sum_{t(i)\in U(n)}\Big|f\big(t(i+1)\big)-f\big(t(i)\big)\Big|\Big|g(t(i+1))-g(t(i))\Big|=0$$
?

Last edited: Jan 24, 2005