Abstract
Cauchy's functional equation is the equation
\[ h(x + y) = h(x) + h(y). \]
This entry formalizes properties of the solutions \(h : \mathbb{R} \to \mathbb{R}\) of Cauchy's functional equation, including:
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A function \(h\) satisfying Cauchy's functional equation has the form
\(h(x) = cx\) for some \(c\) if one of the following conditions holds:
- \(h\) is continuous at some point.
- \(h\) is continuous.
- \(h\) is bounded on some interval.
- \(h\) is (anti)monotone on some interval.
- \(h\) is measurable.
- There are non-linear solutions to Cauchy's functional equation.
- The set of non-linear solutions has cardinality \(|\mathbb{R}^{\mathbb{R}}|\).
- The graph of a non-linear solution\(\{(x, h(x)) \mid x \in \mathbb{R}\}\) is dense in \(\mathbb{R} \times \mathbb{R}\).
License
Note
Subsections “Case: h is Monotone”, “h is Bounded”, “Corollaries”, and section “Non-Linear Solutions”, were written by a generative AI.