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, meaning there are no breaks, jumps, or holes in the graph.

Intervals of Continuity

Intervals of continuity refer to the ranges of the independent variable (usually x) where a function remains continuous. These intervals can be expressed in interval notation, indicating where the function does not have any discontinuities, such as vertical asymptotes or removable discontinuities.

Identifying Discontinuities

Discontinuities can occur in various forms, including removable (holes), jump, and infinite discontinuities. To determine where a function is continuous, one must identify these points by analyzing the function's behavior, particularly at critical points like where the function is undefined or where it changes direction.