Question

Find the x-values (if any) at which f is not continuous. If there are discontinuities, determine whether they are removable. (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.)

f(x) =  −5x,      x ≤ 6
x2 − 4x + 7,      x > 6

Answer 1

Answer: DNE

Explanation:

To find the x-values at which f is not continuous, we need to consider the two parts of the function separately.

For the first part, when x ≤ 6, the function is f(x) = -5x. This is a linear function and is continuous for all real values of x. There are no discontinuities in this part of the function.

For the second part, when x > 6, the function is f(x) = x^2 - 4x + 7. This is a quadratic function. Quadratic functions are continuous everywhere, so there are no discontinuities in this part of the function.

Since there are no discontinuities in either part of the function, f is continuous for all real values of x.

Therefore, the x-values at which f is not continuous are DNE (do not exist).