Calculus How To

Infinite Discontinuity: Definition, Examples

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Types of Discontinuity >

What is an Infinite Discontinuity?

An infinite discontinuity has one or more infinite limits—values that get larger and larger as you move closer to the gap in the function. An infinite discontinuity is a subtype of essential discontinuities, which are a broad set of badly behaved discontinuities that cannot be removed.

types of discontinuity

Types of Discontinuity. The infinite discontinuity on the right has function values that keep on going towards infinity.


one sided infinity

This function only tends to infinity at x = 0 on one side (the right), but it’s still classified as an infinite discontinuity.

It’s important to note that only one side has to tend to ±infinity in order for the discontinuity to be classified as infinite. One side may reach a certain function value, or be undefined. But as long as one side is either negative infinity or positive infinity, then it’s an infinite discontinuity.

Infinity can be Positive or Negative

The function can go towards infinity in the same direction, or in different directions. For example, the function can go towards:

  • Positive infinity on both sides,
  • Negative infinity on both sides, or
  • One side can go to negative infinity and the other towards positive infinity.

graph of 1/x

Graph of y = 1/x, which tends towards both negative and positive infinity at x = 0.



graph of 1 over x squared

Graph of 1/x2, which tends towards negative infinity in both directions at x = 0. (Graphs made with the Desmos Calculator)



References

Infinite Discontinuity. Retrieved October 29, 2019 from: http://www-math.mit.edu/~djk/18_01/chapter02/example03.html
Infinite Discontinuities. Retrieved October 28, 2019 from: https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/part-a-definition-and-basic-rules/session-5-discontinuity/MIT18_01SCF10_Ses5c.pdf
Desmos Calculator.

CITE THIS AS:
Stephanie Glen. "Infinite Discontinuity: Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/infinite-discontinuity-definition-examples/
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