Left and right limits and discontinuities
Let
Discontinuities
Note that we require both ① limits to exist and ② be equal to each other and
③ to
- Asymptotic discontinuity and oscillatory discontinuity
- Jump discontinuity
- Removable discontinuity
- Domain break discontinuity, with function continuity maintained (I made up this name). Not 100% if it exists.
These are discussed further on the back side.
Example
1. A left or right limit doesn't exist
Two examples of discontinuites that arise from a limit not existing:
- Asymptotic discontinuity
- One or both limits do not exist as, informally,
tends to infinity as . Example: has a discontinuity at 0. - Oscillatory discontinuity
- A function can remain bounded but still not have a left or right limit
near a point. For example:
where is if or otherwise.
2. Jump discontinuity
Where both limits exist but are not equal, we have a
jump discontinuity. At least one must not be equal
3. Removable discontinuity
Let an right limits can exist and be equal to
Then both left and right limits exist and we have
4. (Not sure what called?) Domain break discontinuity
If
Considering this example, continuity might best be thought of as the question
of whether for all sequences