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 interest on the graph, which is (5, 0).
2. Observe the behavior of the function around this point. The function starts at (5, 0) and continues to the right.
3. Determine if there are any breaks, holes, or jumps in the graph. In this case, there are none.
4. Conclude that the function is continuous from x = 5 to positive infinity.
5. Express the interval of continuity in interval notation: [5, ∞).

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.