A real-valued function is defined on an interval containing a point
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A real-valued function f is defined on an interval I containing a point c. Answer the following questions about the relationship between differentiability and continuity at the point c.
(i) State the formal (limit) definition of the derivative f′(c).
(ii) Using the definition stated in part (i), prove that if f′(c) exists, then f is continuous at c.
(iii) Explain, with reference to your proof in part (ii), why the converse statement is not always true, citing a standard counter-example from elementary calculus.