Complete the following steps for each function.


c. State the interval(s) of continuity.


f(x)={x^3+4x+1 if x≤0
2x^3 if x>0; a=0

Key Concepts

Continuity of Functions

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 crucial for determining where a function does not have breaks, jumps, or asymptotes.

Piecewise Functions

Piecewise functions are defined by different expressions based on the input value. In this case, the function f(x) has two distinct expressions depending on whether x is less than or equal to zero or greater than zero. Understanding how to analyze each piece separately is essential for determining the overall continuity of the function.

Limits

Limits are fundamental in calculus for understanding the behavior of functions as they approach specific points. To assess continuity at the boundary point (x=0) of the piecewise function, one must evaluate the left-hand limit and the right-hand limit. If both limits exist and are equal to the function's value at that point, the function is continuous there.