A function is defined on the real numbers with the following properties: •
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
A function g is defined on the real numbers with the following properties:
• g(2)=6.
• x→2−limg(x)=5.
• x→2+limg(x)=7.
Answer the following questions related to the behavior of g at x=2.
(a) State the three formal criteria that must simultaneously hold for a function to be continuous at x=2. Explain the meaning of each criterion in the context of limits and function values.
(b) Using the given information, determine which of the three criteria are satisfied and which are not. Provide clear reasoning for each determination.
(c) Based on your work in part (b), classify the discontinuity of g at x=2 (e.g., removable, jump, infinite). Justify your classification by referencing the relationship between the one-sided limits and the function value.