A function is defined piecewise by h(x)= cases x x , & x 0, 2, & x=0
MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability
Problem
A function h is defined piecewise by
h(x)=⎩⎨⎧xsinx,2,x=0,x=0.
Answer the following questions about the point x=0.
(1) Part 1: (a) State the three conditions that must simultaneously hold for a real-valued function to be continuous at a point.
(2) Part 2: (b) Verify whether each of the three conditions is satisfied for h at x=0, showing all reasoning.
(3) Part 3: (c) Based on your work in part (b), explain in complete sentences which condition(s) fail and why this leads to the conclusion that h is or is not continuous at x=0. Include appropriate mathematical justification in your explanation.