MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Consider the function F(x)=∣x∣.
(i) State the definitions of continuity and differentiability of a real‐valued function at a point.
(ii) Verify whether F(x) is continuous at x=0.
(iii) Determine whether F(x) is differentiable at x=0. Give full justification based on the behaviour of the relevant limits.
(iv) Interpret geometrically your conclusion in part (iii) with reference to the graph of y=∣x∣. What does this tell you about the tangent at the origin?