I know that $f(x,y) = (x^2-y^2,xy)$ is not lipschitz. Maybe it would be locally lipschitz, then I think apply standard argument that for each bounded measure zero set is measure zero, and extend it to unbounded set, and for any positive measure set, take $G_{\delta}$ set which differ from measure zero, then send it using continuity. But I think this may have problem that if lipschitz constant increase faster than my expectation, then the argument cannot hold.
So my question is this; is there any hint to show that it preserves measure zero set as measure zero?