yes, it's not smooth by definition

   d/dx √(x²) = x/√(x²)

because for x=0 the derivative is 0/0, but in this case it might be interpreted as the interval [-1,1]. That's just the way it is often plotted. I come up short with an algebraic explanation, it might as well be ]-∞,∞[ (deriving it from z(x,y)=1/y for example), but I imagine this as a bundle of tangents on the origin, parameterized by the interval. That's an infinitesimal curve, not just an infinitesimal slope if you will. So it's not linear algebra (I guess). The integral of the derivative is obviously defined everywhere, explain that. It's zero at zero, because the sum of the interval is zero. And if we are only interested in the boundaries of the interval at 0, than that's the tuple (-1,1). Just like complex numbers or vectors are tuples (but this isn't a complex number I guess).