MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
The function g(x)=∣x∣ is continuous for all real x. Which statement best explains why g is not differentiable at x=0?
(a) g is not differentiable at x=0 because the function itself is not continuous there.
(b) In the difference‐quotient formula x−0∣x∣−∣0∣, setting x=0 forces division by 0, so the derivative cannot be evaluated.
(c) The limits of the difference quotient from the left and right at x=0 are unequal.
(d) At x=0 the sharp turn makes both one–sided slopes 0, but a cusp is never differentiable.