Math 2.0

Real functions Limits and continuity Concept

Classification of discontinuities

Builds on Continuity at a pointOne-sided limit

Summary

A point of the domain where a real function is not continuous is a discontinuity. Discontinuities are sorted by the behavior of the one-sided limits at the point.

Remarks

At a removable discontinuity, the limit exists but differs from the value. At a jump discontinuity, both one-sided limits exist but they differ. At an essential discontinuity, at least one one-sided limit does not exist. An example is sin⁡(1/x)\sin(1/x) at 00, with any value at 00. A monotone function has countably many discontinuities, and all of them are jumps.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionClassification of discontinuities, concept, no evidence levelClassification of dis…concept
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Limit of a function definition
  3. Continuity at a point definition
  4. One-sided limit definition
  5. Classification of discontinuities concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Classification_of_discontinuities, Classification
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Tag04R9K5H562
URIm2://04R9K5H562
Kindconcept
WritingW1 stated
MSC26A15
UsesContinuity at a point, One-sided limit
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