The function is defined for by the following verbal description of its graph
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
The function f is defined for −1≤x≤3 by the following verbal description of its graph.
• For −1≤x<0, the graph is a straight line segment with slope −1 that passes through the point (−1,0).
• At x=0 the point (0,0) is solid (included in the graph).
• For 0<x≤3, the graph is the right half of the curve y=x translated 1 unit upward.
• At x=3 the point (3,2) is shown as an open circle (not included in the graph).
Answer the following questions about f.
(a) Determine whether f is continuous at x=0 and at x=3. Give a brief explanation for each point.
(b) Determine whether f is differentiable at x=0 and at x=3. Clearly specify the reason in each case.
(c) Describe the relationship between your answers in parts (a) and (b). Explain how the results illustrate the general connection between continuity and differentiability.