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

Verified step by step guidance

1. Step 1: Identify the points where the function might be discontinuous by looking for breaks, holes, or jumps in the graph. In this graph, there is a clear discontinuity at the point (2, 0).
2. Step 2: Understand that a function is continuous on an interval if you can draw the graph without lifting your pencil. The graph is continuous on intervals where there are no breaks or holes.
3. Step 3: Observe the graph to the left of x = 2. The curve is smooth and unbroken from negative infinity up to but not including x = 2, so the function is continuous on the interval \((-\infty, 2)\).
4. Step 4: Observe the graph to the right of x = 2. The curve is also smooth and unbroken from just after x = 2 to positive infinity, so the function is continuous on the interval \((2, \infty)\).
5. Step 5: Conclude that the function is continuous on the intervals \((-\infty, 2)\) and \((2, \infty)\), but not continuous at x = 2 due to the discontinuity shown by the break in the graph.

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. Understanding the domain is essential to determine where the function exists and to analyze continuity only within those valid input values.

Types of Discontinuities

Discontinuities occur where a function is not continuous. Common types include jump discontinuities, removable discontinuities (holes), and infinite discontinuities (vertical asymptotes). Identifying the type helps in understanding the behavior of the function at specific points, such as the discontinuity at x=2 in the graph.