Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05
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A singular function on a compact interval
Definition
Let .
We call a singular function on when:
- is continuous (Continuity of at a point of and on : the - condition, its agreement with at a limit point, and continuity at an isolated point);
- is nondecreasing (Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of , with the dictionary to monotone sequences);
- is not constant;
- exists almost everywhere in the sense of The derivative of at a point that is a limit point of , and differentiability on a set and equals almost everywhere.
Remarks
- This is the function-level form of a singular continuous measure: the increase is carried by a null set.
- The Cantor function is the canonical example, and later examples build other singular functions from it.
Depends on
- Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point
- The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set
- Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length
- Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $\mathbb{R}$, with the dictionary to monotone sequences
Used by
- A strictly increasing singular function from a dense series of scaled Cantor functions Example
- FALSE: every increasing function satisfies Newton-Leibniz with its derivative False statement
- A right-continuous nondecreasing function splits uniquely as absolutely continuous plus jump plus singular continuous Theorem
Dependency tree · two levels
18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- A. M. Bruckner, J. B. Bruckner, and B. S. Thomson, Real Analysis, 2nd ed. (standard reference, not scraped)