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.

Identifying Discontinuities

Discontinuities occur where a function is not continuous. Common types include removable discontinuities (holes), jump discontinuities (jumps in the graph), and infinite discontinuities (asymptotes). In the provided graph, the point (2, 0) indicates a removable discontinuity, as the function approaches a value but is not defined at that point.

Intervals of Continuity

Intervals of continuity refer to the ranges of x-values over which a function remains continuous. To determine these intervals, one must analyze the graph for any points of discontinuity and then express the continuous sections using interval notation, excluding points where the function is not defined.