Determine the intervals of the domain over which each function is continuous. See Example 1.

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.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. Understanding the domain is essential for analyzing continuity, as discontinuities often arise from values that are not included in the domain, such as division by zero or taking the square root of a negative number.

Types of Discontinuities

Discontinuities can be classified into three main types: removable, jump, and infinite. A removable discontinuity occurs when a function is not defined at a point but can be made continuous by redefining it. Jump discontinuities occur when there is a sudden change in function value, while infinite discontinuities happen when the function approaches infinity at a certain point. Identifying these types helps in determining the intervals of continuity.