Which of the following functions are discontinuous?I) f(x) = x^2 + 1 when x < 0 and f(x) = x + 1 when x ≥ 0II) f(x) = x when x < 0, f(x) = 1 when x = 0, and f(x) = -x when x > 0III) f(x) = 2

To determine if a function is discontinuous, we need to check if there is any point in the domain of the function where the function is not defined, or where the limit does not exist, or where the limit exists but is not equal to the function's value at that point.

I) f(x) = x^2 + 1 when x < 0 and f(x) = x + 1 when x ≥ 0

At x = 0, the function is defined and the limit from both sides exists. The limit from the left is (0)^2 + 1 = 1 and the limit from the right is 0 + 1 = 1. Since the limit from both sides is equal to the function's value at x = 0, the function is continuous at x = 0. Therefore, this function is continuous everywhere.

II) f(x) = x when x < 0, f(x) = 1 when x = 0, and f(x) = -x when x > 0

At x = 0, the function is defined and the limit from both sides exists. The limit from the left is 0 and the limit from the right is also 0. However, the function's value at x = 0 is 1, which is not equal to the limit from either side. Therefore, this function is discontinuous at x = 0.

III) f(x) = 2

This function is defined for all x and the limit exists and is equal to the function's value for all x. Therefore, this function is continuous everywhere.

So, the function that is discontinuous is II) f(x) = x when x < 0, f(x) = 1 when x = 0, and f(x) = -x when x > 0.