Question

In Exercises \(1-10,\) find the points of continuity and the points of discontinuity of the function. Identify each type of discontinuity. $$y=\sqrt[3]{2 x-1}$$

Short answer

The given function \(y=\sqrt[3]{2x-1}\) is continuous for all real values of x, and hence doesn't have any points of discontinuity or any type of discontinuity.

Step-by-step solution

Determine the Domain of the function

The function contains a cube root, which can handle any real number input. So the domain of this function is \(x \in (-\infty, \infty)\).

Check for Continuity

Since a cube root function is continuous for all values in its domain, we can say that the function \(y=\sqrt[3]{2x-1}\) is continuous for all real numbers.

Check for points of Discontinuity

A function is discontinuous at points not included in its domain. However in this case, since the domain covers all real numbers, the function has no points of discontinuity.

Identify type of discontinuity

Once we have established that there are no points of discontinuity, we can safely conclude that the function doesn't have any type of discontinuity since it is continuous everywhere within its domain.