A claim is made: “If a function is continuous at a point
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A claim is made: “If a function is continuous at a point a, then it must also be differentiable at a.”
(a) Indicate whether the claim is always true or sometimes false. Provide a brief explanation of your decision.
(b) Construct an explicit function f and a real number c such that f is continuous at c but not differentiable at c.
(c) Prove rigorously that the function you provided in part (b) is continuous at c.
(d) Prove rigorously that the function you provided in part (b) is not differentiable at c. Be sure to address the appropriate one–sided limits of the difference quotient.