Question: is mod x differentiable at 0?
Answer: ⎢⎝ x x > 0 0 x = 0 −x x < 0 . ... The left limit does not equal the right limit, and therefore the limit of the difference quotient of f(x) = |x| at x = 0 does not exist. Thus the absolute value function is not differentiable at x = 0.