Combining (block)-sensitivity and Lipschitz conditions?
One possible definition for a given point and ϵ would be the following: perturb each coordinate of the input independently with a Gaussian with mean 0 and variance $\epsilon^2/n$ and then look at the variance of the function's value with respect to this input distribution. It's also possible to perturb a specific coordinate this way and keep the rest fixed, depending on your application.

You may also find this paper of interest: "Noise stability of functions with low influences: invariance and optimality" by E. Mossel, R. O'Donnell, K. Oleszkiewicz.