A mathematical function that is linear but not continuous
In short: Most people assume linear functions must be straight lines. This video explores a bizarre mathematical construction that satisfies the rules of linearity while remaining completely disconnected.
Real numbers should be perceived as a vector space defined over the rational numbers to grasp this concept. Using the Axiom of Choice, mathematicians can identify a basis—a subset of real numbers where every real number can be expressed as a linear combination of these basis elements using only rational coefficients.
The function is defined by taking the sum of those rational coefficients. While it strictly follows the rules of linearity, it is impossible to draw as a line because its output is always a rational number, meaning it misses almost every real value and fails the test of continuity.
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Sources
- Summary by Intellect · source: The strangest function I know (linear but not continuous?? huh??)
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