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

Verified step by step guidance

1. Step 1: Understand the concept of continuity for a function. A function is continuous at a point if the limit of the function as it approaches the point from both sides equals the function's value at that point.
2. Step 2: Observe the graph carefully. Notice that the function is a straight line except at the point (1, 13), where there is an open circle, indicating the function is not defined at that point.
3. Step 3: Since the function is a line everywhere else, it is continuous on all intervals except possibly at x = 1. Check the value of the function at x = 1 by looking at the graph or given points.
4. Step 4: Because the function is not defined at x = 1 (open circle), it is not continuous at x = 1. However, the function is continuous on the intervals to the left and right of x = 1.
5. Step 5: Conclude that the function is continuous on the intervals \((-\infty, 1)\) and \((1, \infty)\), but not continuous at \(x = 1\).

Key Concepts

Continuity of a Function

A function is continuous at a point if the limit of the function as it approaches the point equals the function's value at that point. This means there are no breaks, jumps, or holes in the graph at that point. Continuity over an interval means the function is continuous at every point within that interval.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. When determining continuity, it is important to consider only the domain values where the function exists, as continuity cannot be evaluated outside the domain.

Interpreting Graphs for Continuity

Graphs visually show where a function is continuous or discontinuous. Open circles indicate points where the function is not defined or has a different value, signaling discontinuity. A continuous function graph has no breaks or holes, so identifying these features helps determine intervals of continuity.