Alphabeta Math

Monotone Functions, Discontinuities, and Continuity Sets: Examples and Counterexamples

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1 · Prerequisites
Abstract Algebra
Linear Algebra
Number Theory
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

Thomae's function computed: t(1/2) = 1/2, t(2/3) = 1/3, t(m) = 1 at every integer m, t(x) = 0 at every irrational, and ωₜ(c) = t(c) at every real c

Example

A bounded nondecreasing f : ℝ → ℝ whose set of discontinuities is exactly ℚ, obtained from the prescribed-jump construction applied to one fixed enumeration of the rationals

Example

Froda's countable bound is attained: a bounded nondecreasing function on ℝ discontinuous exactly at the points 1 - 1/(k+1) for k ∈ ℕ, an infinite discontinuity set inside a bounded interval

Example

The n-th root as a continuous inverse: for a natural n ≥ 1 the map x ↦ xⁿ is continuous and strictly increasing on [0,∞) with image [0,∞), so its inverse x ↦ x^1/n is continuous and strictly increasing

Example

The Cantor set has measure zero, yet the Cantor function maps it onto all of [0,1]: a null set can have image an interval of length 1

Example

The function equal to q at a rational p/q in lowest terms and to 0 at every irrational is finite at every point and unbounded on every nondegenerate interval

Example

The Dirichlet function is the pointwise limit of a sequence of Baire class one functions and is itself not Baire class one, so the Baire hierarchy on [0,1] is already strict at the first level

Example

An additive f : ℝ → ℝ that is not x ↦ cx: the coefficient of one fixed Hamel basis vector. It is unbounded above and below on every nondegenerate interval, its graph is dense in ℝ², and every nonempty level set is dense in ℝ

Example

A bounded function on ℝ with no local maximum and no local minimum at any point, upper semicontinuous at no point and lower semicontinuous at no point: compose the Hamel coefficient with a strictly increasing injection of ℝ into (0,1)

Counterexample

An upper semicontinuous function on [0,1] that is bounded below and attains no minimum, so the semicontinuous extreme value theorem is genuinely one-sided

Counterexample

A continuous injection on [0,1] ∪ [2,3] that is not monotone, so the interval hypothesis cannot be dropped from the strict-monotonicity theorem

Sources