I have never seen anyone do this but has anyone defined the variance for a countinous function $\displaystyle f(x)$ on interval $\displaystyle [a,b]$?

I used "reasonable" definition to get:

$\displaystyle \frac{1}{b-a}\int^b_a[f(x)-\= x]^2dx$

where,

$\displaystyle \=x$ is the average value, thus,

$\displaystyle \=x=\frac{1}{b-a}\int^b_a f(x)dx$