How do I know if a graph has removable continuity, jump, or infinite? What is the difference?

The only removable one is a point discontinuity, in which a single point on the graph is missing. The other two discontinuities, jump and infinite, are nonremovable. A jump discontinuity is when the graph "jumps" in y-value and does not connect, while an infinite discontinuity is an asympotote.

Determining if a graph has removable continuity, a jump, or infinite behavior involves analyzing the behavior of the function at specific points. Let's explore each concept:

1. Removable continuity: A function has removable continuity at a specific point if there is a hole or gap in the graph, but the function can be made continuous by defining the value of the function at that point. To determine whether a graph has removable continuity at a specific point, follow these steps:
a. Evaluate the function at the point in question. If the function is undefined, go to step b. Otherwise, the graph does not have removable continuity at that point.
b. If the function is undefined at the point, see if you can define it in a way that makes the function continuous. This might involve using limits or algebraic simplification. If you can define the function at the point so that it becomes continuous, the graph has removable continuity at that point.

2. Jump Discontinuity: A function exhibits a jump discontinuity at a specific point when the left-hand limit and the right-hand limit of the function exist, but they are not equal. To identify a jump discontinuity, follow these steps:
a. Find the left-hand limit and the right-hand limit of the function at the point in question.
b. If the left-hand limit and the right-hand limit exist but are not equal, a jump discontinuity is present. The graph will often exhibit a vertical jump or gap at that point.

3. Infinite Discontinuity: A function has an infinite discontinuity at a specific point if at least one of the left-hand or right-hand limits evaluates to positive or negative infinity. To recognize an infinite discontinuity, perform the following steps:
a. Calculate the left-hand limit and the right-hand limit of the function at the point in question.
b. If either the left-hand limit or the right-hand limit evaluates to positive or negative infinity, the graph has an infinite discontinuity. This typically manifests as a vertical asymptote, where the function approaches infinity or negative infinity.

In summary, removable continuity involves a gap that can be filled, jump discontinuity has unequal left-hand and right-hand limits, and infinite discontinuity occurs when at least one limit leads to positive or negative infinity. By evaluating the function at a specific point and analyzing the behavior of the left-hand and right-hand limits, you can determine which type of discontinuity is present in a graph.