As above, if we have a 1-Lipschitz function $f$, does that imply:
$$I(f(X); f(Y)) = I(X; Y)$$
for some random variables $X$ and $Y$?
I think the main thing is to show $f$ is injective, and use the fact that mutual information is preserved under invertible transformations. But I don't know how to show a 1-Lipschitz function is also an injective one, although it seems intuitive to be so.
Is there a way to prove this?