Question

determine the points in the interval \\((1, 6)\\) at which the function has discontinuities. for each point state the conditions in the continuity checklist that are violated. in order for \\(f\\) to be continuous at \\(a\\), the following three conditions must hold:

1. \\(f(a)\\) is defined (\\(a\\) is in the domain of \\(f\\)).
2. \\(\lim_{x \to a} f(x)\\) exists.
3. \\(\lim_{x \to a} f(x) = f(a)\\) (the value of \\(f\\) equals the limit of \\(f\\) at \\(a\\)).

\\(f(x)\\) is discontinuous at \\(x = 2, 3, 4\\)
(use a comma to separate answers as needed.)

select the correct choice below and, if necessary, fill in the answer box within your choice.

a. the point(s) at \\(x =\\) violate(s) the first condition in the continuity checklist.
(use a comma to separate answers as needed.)
b. no points violate the first condition in the continuity checklist.

Explanation:

Identify points of discontinuity in the interval \( (1, 6) \)

From the graph, the function \( f(x) \) is discontinuous at:

• \( x = 2 \) (vertical asymptote / infinite discontinuity)
• \( x = 3 \) (removable discontinuity / hole)
• \( x = 4 \) (jump discontinuity)

Evaluate the first condition of continuity at each point

The first condition states that \( f(a) \) must be defined:

• At \( x = 2 \): There is a vertical asymptote, so \( f(2) \) is undefined.
• At \( x = 3 \): There is a hole at \( (3, 2) \) and no solid point at \( x = 3 \), so \( f(3) \) is undefined.
• At \( x = 4 \): There is a solid point at \( (4, 1) \), so \( f(4) = 1 \) is defined.

Thus, the points that violate the first condition are \( x = 2, 3 \).

Select the correct multiple-choice option

Since there are points that violate the first condition (\( x = 2, 3 \)), Option A is correct, and the values to fill in are \( 2, 3 \). Option B is incorrect.

Answer:

• A. The point(s) at x = 2,3 violate(s) the first condition in the continuity checklist (Correct answer)
• B. No points violate the first condition in the continuity checklist.