Math 2.0

Real functions Definition

Jump discontinuity

Builds on One-sided limit

Definition

Let D⊆ℝD \subseteq \mathbb{R}, let f ⁣:D→ℝf \colon D \to \mathbb{R}, and let a∈Da \in D be a limit point of both D∩(−∞,a)D \cap (-\infty, a) and D∩(a,∞)D \cap (a, \infty). The point aa is a jump discontinuity of ff if both one-sided limits lim⁡x→a−f(x)\lim_{x \to a^-} f(x) and lim⁡x→a+f(x)\lim_{x \to a^+} f(x) exist and are different.

Remarks

The difference of the two one-sided limits is the size of the jump. For example, the sign function has a jump discontinuity at 00, of size 22. Monotone functions have only jump discontinuities.

Reading path

To understand this page from the ground up, read these 4 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 functiondefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionJump discontinuity, definition, no evidence levelJump discontinuitydefinition
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  • no evidence level
  1. Limit point of a set definition
  2. Limit of a function definition
  3. One-sided limit definition
  4. Jump discontinuity definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Classification_of_discontinuities, Jump discontinuity
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DefinesJump discontinuity
UsesOne-sided limit

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