Infinitesimal independence Let's say we have two random variables $X$ and $Y$. 
Is there a name for saying that $X$ and $Y$ are independent only for the values concentrated around a small interval around some $x_0$ and $y_0$ ? Like: 
\begin{equation}
\textrm{For all}~(x, y) \in [x_0 - \epsilon, x_0 + \epsilon] \times [x_1 - \epsilon, x_1 + \epsilon]: \\P(X=x,Y=y) = P(X=x)\,P(Y=y)
\end{equation}
Or maybe we could say that some sort of local mutual information of $X$ and $Y$ in $x_0$ and $y_0$ is null ?  
 A: In fact I realized my definition with the small interval around points was weird: I realized there exist the Pointwise Mutual Information (https://en.wikipedia.org/wiki/Pointwise_mutual_information), that will simply measure the independence between two random variables at a particular point x, y. I guess it was what I was looking for.  
A: If the $X$ and $Y$ rv's have a joint density $f(x,y)$, then you can compute the marginals of $X$ and $Y$, say $g(x), h(y)$ and compare the joint density to the independence density $g(x)h(y)$, for example via the likelihood ratio 
$$
   \frac{f(x,y)}{g(x)h(y)}
$$
and plot that. In the following we have done this for the example of a joint Cauchy distribution:

The R code used is:
library(mvtnorm)
library(lattice)

mycplot <- function(df=1, ...) {
    x <- y <- seq(from=-3,  to=3,  length.out=101)
    grid <- expand.grid(x=x, y=y)
    grid$z <- exp( mvtnorm::dmvt(cbind(grid$x, grid$y), df=df, log=TRUE) -
                   dt(grid$x, df=df, log=TRUE) - dt(grid$y, df=df, log=TRUE))
    lattice::contourplot(z ~ x*y, data=grid, region=TRUE, ...)
}

