Limits and Continuity
On what intervals are the following functions continuous?


a. ƒ(x) = x¹/³

Verified step by step guidance

1. Understand the concept of continuity: A function is continuous at a point if the limit of the function as it approaches the point from both sides is equal to the function's value at that point. For a function to be continuous on an interval, it must be continuous at every point within that interval.
2. Identify the function: The given function is \( f(x) = x^{1/3} \). This is a cube root function, which is a type of root function.
3. Analyze the domain of the function: The cube root function \( x^{1/3} \) is defined for all real numbers because you can take the cube root of any real number, whether positive, negative, or zero.
4. Determine the intervals of continuity: Since the cube root function is defined for all real numbers and does not have any points of discontinuity (such as holes, jumps, or vertical asymptotes), it is continuous everywhere on its domain.
5. Conclude the intervals of continuity: Therefore, the function \( f(x) = x^{1/3} \) is continuous on the entire set of real numbers, which can be expressed as the interval \((-\infty, \infty)\).