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


b. g(x) = x³/⁴

Verified step by step guidance

1. Identify the type of function: The function g(x) = x^(3/4) is a power function, which is generally continuous wherever it is defined.
2. Determine the domain of the function: Since the exponent 3/4 is a positive rational number, g(x) is defined for all x ≥ 0. This is because the fourth root of a negative number is not real.
3. Check for any points of discontinuity: For power functions like g(x) = x^(3/4), there are no discontinuities within their domain. Therefore, the function is continuous on its entire domain.
4. Conclude the intervals of continuity: Since g(x) is defined and continuous for all x ≥ 0, the interval of continuity is [0, ∞).
5. Summarize the findings: The function g(x) = x^(3/4) is continuous on the interval [0, ∞), as it is a power function with a domain of all non-negative real numbers.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. Understanding limits is crucial for analyzing the behavior of functions, especially at points where they may not be explicitly defined. They help determine continuity and differentiability, which are essential for evaluating functions over intervals.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval. This concept is vital for understanding how functions behave and ensuring that there are no breaks, jumps, or holes in their graphs.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals. Analyzing these functions requires checking continuity at the boundaries of the pieces to ensure that the function behaves consistently. Understanding how to evaluate these functions across their defined intervals is essential for determining where they are continuous.