[sfpotter]

This is a great point... and Rademacher's theorem is one of my favorites. But this is in the context of a numerical algorithm which proposes to replace interval arithmetic with a direct evaluation of a gradient. For an SDF, the gradient fails to exist in many places; and since we're working with it on a computer and not doing math, this algorithm will eventually evaluate the gradient at a point where it is undefined (finite number of points, p > 0). On the other hand, it's straightforward to come up with a useful interval inclusion for the gradient which is defined even in these places (it should contain the subgradient of the function at each point). So, I am personally not convinced of the value of the proposed approach.

[qbit42]

Yeah in context I somewhat agree, though the utility for graphics applications probably comes down to some more empirical aspects that I won't conjecture about. I imagine there is some stuff you could do in this setting by incorporating autograd derivatives from many slightly perturbed points in a neighborhood of the input point (which together act as a coarse approximation of the subdifferential set).