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

Verified step by step guidance

1. Identify the point of discontinuity in the graph. Here, the function is not defined at x = 2.
2. Observe the behavior of the function on either side of the discontinuity. The function approaches different values as x approaches 2 from the left and right.
3. Determine the intervals where the function is continuous. The function is continuous on the intervals where it does not include the point of discontinuity.
4. Write the intervals of continuity. In this case, the function is continuous on the intervals (-∞, 2) and (2, ∞).
5. Combine the intervals to express the domain over which the function is continuous. The function is continuous on (-∞, 2) ∪ (2, ∞).

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.