MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A function f is defined by
f(x)={∣x∣,x2,x≤0,x>0.
Answer the following questions about the behavior of f at x=0.
(1) Part 1: (a) Determine whether f is continuous at x=0. Show the necessary limit calculations and state your conclusion.
(2) Part 2: (b) Determine whether f is differentiable at x=0. Evaluate the appropriate one-sided limits and justify your answer.
(3) Part 3: (c) Explain how your results in parts (a) and (b) illustrate the relationship between continuity and differentiability at a point.