Removable Discontinuity
Removable discontinuity, also known as a removable singularity, occurs when a function is not continuous at a specific point due to the presence of a hole or a gap in the graph. This type of discontinuity can be removed by redefining the function at that point. A removable discontinuity is characterized by the following properties:- The function approaches the same limit as x approaches the discontinuous point from both sides.
- The function has a hole or gap in the graph at the discontinuous point.
- The function can be made continuous by redefining it at the discontinuous point.
Infinite Discontinuity
- The function approaches positive or negative infinity as x approaches the discontinuous point.
- The function is not defined at the discontinuous point.
Essential Discontinuity
Essential discontinuity occurs when a function approaches different limits as x approaches a specific point from different sides. This type of discontinuity is characterized by the following properties:- The function approaches different limits as x approaches the discontinuous point from different sides.
- The function is not defined at the discontinuous point.
Jump Discontinuity
Jump discontinuity occurs when a function has a sudden jump or gap in its graph at a specific point. This type of discontinuity is characterized by the following properties:- The function has a sudden jump or gap in its graph at the discontinuous point.
- The function approaches different values on either side of the discontinuous point.
Table of Discontinuity Types
| Discontinuity Type | Characteristics | Example |
|---|---|---|
| Removable | Function approaches the same limit from both sides, has a hole or gap in the graph | f(x) = (x^2 - 4) / (x - 2) |
| Infinite | f(x) = 1 / (x - 1) | |
| Essential | Function approaches different limits from different sides | f(x) = |x - 1| / (x - 1) |
| Jump | Function has a sudden jump or gap in the graph | f(x) = |x - 1| |
- Plot the graph of the function to visualize the discontinuity.
- Check if the function approaches the same limit from both sides of the discontinuous point.
- Check if the function approaches positive or negative infinity.
- Check if the function has a sudden jump or gap in the graph.