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


d. k(x) = x⁻¹/⁶

Verified step by step guidance

1. First, recall the definition of continuity for a function at a point. 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.
2. Consider the function k(x) = x^{-1/6}. This function involves a fractional exponent, which can be problematic at certain points, particularly where the base of the exponent is zero or negative.
3. Identify the domain of the function k(x). Since x^{-1/6} involves taking the sixth root of x, the function is undefined for x = 0, as division by zero is not possible.
4. Additionally, consider the behavior of the function for negative values of x. The sixth root of a negative number is not defined in the real number system, so the function is also undefined for x < 0.
5. Conclude that the function k(x) = x^{-1/6} is continuous on the interval (0, ∞), as it is defined and continuous for all positive values of x.