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

Verified step by step guidance

1. Step 1: Identify the points where the function might be discontinuous. In the graph, there is a visible discontinuity at the point (2, 0), where the function has a break or jump.
2. Step 2: Understand that a function is continuous on an interval if there are no breaks, jumps, or holes in the graph within that interval. So, we need to look at the graph on either side of x = 2.
3. Step 3: Observe the graph to the left of x = 2. The function appears smooth and unbroken from negative infinity up to but not including x = 2. This means the function is continuous on the interval \((-\infty, 2)\).
4. Step 4: Observe the graph to the right of x = 2. The function is also smooth and unbroken from just after x = 2 to positive infinity. This means 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 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 from both sides equals the function's value at that point. Continuity over an interval means the function has no breaks, jumps, or holes within that interval.

Discontinuity and Types

Discontinuities occur where a function is not continuous. Common types include jump discontinuities, removable discontinuities (holes), and infinite discontinuities. Identifying these helps determine where the function fails to be continuous.

Domain and Interval Notation

The domain of a function is the set of all input values for which the function is defined. Interval notation is used to express continuous intervals within the domain, excluding points of discontinuity to clearly specify where the function is continuous.