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. From the graph, observe the point at \(x = 2\) where there is a filled dot at \((2, 0)\), indicating a possible discontinuity.
2. Step 2: Recall that a function is continuous at a point \(x = a\) if three conditions are met: (1) \(f(a)\) is defined, (2) the limit \(\lim_{x \to a} f(x)\) exists, and (3) \(\lim_{x \to a} f(x) = f(a)\).
3. Step 3: Check the continuity at \(x = 2\). The graph shows the function approaches the value 0 from both sides as \(x\) approaches 2, and the function value at 2 is also 0, so the function is continuous at \(x = 2\).
4. Step 4: Since the graph is a smooth curve with no breaks or holes elsewhere, the function is continuous on all other intervals except possibly at points not shown as discontinuous.
5. Step 5: Conclude that the function is continuous on the entire real line \((-\infty, \infty)\), including at \(x = 2\), because the graph shows no breaks, jumps, or holes.

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 can be evaluated and to analyze continuity within those intervals.

Types of Discontinuities

Discontinuities occur when a function is not continuous at a point. Common types include removable discontinuities (holes), jump discontinuities (sudden jumps in value), and infinite discontinuities (vertical asymptotes). Identifying the type helps in understanding the behavior of the function at those points.