Math 2.0

concept

Classification of discontinuities

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.

Explanations

L1 human

A function can fail to be continuous in different ways. A hole in the graph can be repaired, a jump cannot, and some functions oscillate too wildly to have any limit.

What this node needs for W2
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  • edge instance_of in: at least two examples (0 of 2)
  • edge not_instance_of in: at least one non-example (0 of 1)
  • edge specializes both: the more general notions and the special cases (0 of 1)

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To read this node, read these 5 nodes in this order. The foundations come first, and the list ends with this node.

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
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Classification_of_discontinuities, Classification

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