sgn(x): Discontinuous at x = 0. Not differentiable at x = 0. |x|: Continuous everywhere. Not differentiable at x = 0. |x|' = sgn(x) for x != 0. [x]: Discontinuous at all integers. Not differentiable at integers. [x]' = 0 for non-integers. {x}: Discontinuous at integers. {x}' = 1 for non-integers. x|x|: Continuous and differentiable everywhere. (x|x|)' = 2|x|. $x^n$|x|: Differentiable at 0 for n >= 1. |sin x|: Continuous everywhere. Not differentiable at x = n*pi (where sin = 0). |cos x|: Continuous everywhere. Not differentiable at x = (2n+1)*pi/2.