A function is defined on an open interval containing
MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability
Problem
A function g is defined on an open interval containing x=4. You are told that x→4limg(x)=7 and that g(4) is undefined.
(1) Part 1: State the three distinct conditions that must all be satisfied for a function to be continuous at a point x=c. Use precise mathematical language.
(2) Part 2: Explain, with explicit reference to the conditions you stated in part (a), why the information given above is sufficient to conclude that g is not continuous at x=4. Your explanation should clearly cite which condition(s) fail and why.
(3) Part 3: Suppose a new function h is created by defining h(x)=g(x) for x=4 and h(4)=7. Determine whether h is continuous at x=4. Justify your conclusion by verifying each condition of continuity at that point.