[qbit42]

If a function is Lipschitz, then it is differentiable almost everywhere by Radamacher's theorem. Moreover, if you want to prove things about Lipschitz functions, you can often (though not always) prove things about continuously differentiable functions with bounded gradients, and then lift to all Lipschitz functions via an appropriate compactness argument.

[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).

[constantcrying]

Exactly! That is why it is totally superfluous to define Lipschitz continuity in terms of differentiable functions.