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Discontinuous Function

A function that is not continuous function at one or more points in its domain. At these points, the limit of the function does not equal the function’s value, or the limit does not exist.

Key Characteristics

• Limit Mismatch: ${\lim }_{x\to c}f(x)\ne f(c)$
• Visual Breaks: Graphs exhibit jumps, holes, or asymptotes.
• Classification: Includes jump discontinuity, removable discontinuity, and essential discontinuity.

Pathological Examples

Standard discontinuities (e.g., step functions) are intuitive. However, certain functions exhibit extreme discontinuity properties that challenge standard intuition:

• Conway’s Base 13 Function: A function defined on the real numbers that takes on every real value in every interval, no matter how small. It is a classic example of a function that is discontinuous everywhere yet defined everywhere.
• Intuitive Challenge: While standard functions map inputs to unique outputs, this concept challenges the visual representation of functions, often defying the expectation that a function’s graph can be drawn without lifting the pen.
• Educational Resource: For a detailed explanation of its unique mathematical definition and construction, see Conway’s Base 13 Function: Explaining its Unique Mathematical Definition.

References

• Conway’s Base 13 Function: Explaining its Unique Mathematical Definition (Numberphile). https://www.youtube.com/watch?v=zsBaApyf_Ns