Calculus How To

Oscillating Discontinuity

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Types of Discontinuity >

What is an Oscillating Discontinuity?

An oscillating discontinuity (also called an infinitely oscillating discontinuity) jumps about wildly (i.e. oscillates) as it approaches the limit; there’s no way to “repair” the discontinuity. It’s often defined by exclusion: it isn’t a removable discontinuity, a jump discontinuity, or an infinite discontinuity. Therefore, you might see it referred to as an “other” type of discontinuity.


Bounded and Unbounded Functions

Oscillating discontinuities are bounded. In other words, their oscillations stay between certain lines. For example, the function might be bounded between a high point of y = 3 and a low point of y = -3. If the function is unbounded at one or both sides, it’s an infinite discontinuity, even if it’s an oscillating function. In other words, the function must be completely bounded at all points in order for there to be an oscillating discontinuity. In addition, the one sided limits do not exist at all.

Example

Perhaps not surprisingly, many oscillating functions have at least one oscillating discontinuity.

oscillating discontinuity

It’s impossible to pin down a limit for the function sin(1/x) at zero.


Oscillating discontinuities are a sub-type of essential, or non-repairable, discontinuities.

How to Find an Oscillating Discontinuity

The easiest way to identify this type of discontinuity is by continually zooming in on a graph: no matter how many times you zoom in, the function will continue to oscillate around the limit.

On the TI-89, graph the function in a small window (for example, a [-2,2,1]*[-2, 2, 1] window. Press the Zoom key, select the Zoom Box feature (1:ZBox) and press Enter. Mark the upper left of the box by pressing Enter, then enlarge the box using the arrow keys. Press Enter again. Repeat ad infinitum. For full instructions and images of how to do this, see: Module 8: Continuity on the TI website.

References

Grigoriu, M. Stochastic Calculus: Applications in Science and Engineering.

CITE THIS AS:
Stephanie Glen. "Oscillating Discontinuity" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/oscillating-discontinuity/
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