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.

Domain of a Function

The domain of a function is the complete set of possible values (inputs) for which the function is defined. Understanding the domain is crucial for determining where a function is continuous, as discontinuities can occur at points not included in the domain.

Graphical Interpretation

Analyzing the graph of a function provides visual insight into its behavior, including continuity. Points where the graph is not connected or has holes indicate discontinuities, while a smooth, unbroken line suggests continuity across the interval. The graph in the question shows a point at (1, 13) that is open, indicating a discontinuity at that specific point.