Monotone Functions, Discontinuities, and Continuity Sets

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Monotone Functions, Discontinuities, and Continuity Sets

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Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of ℝ, with the dictionary to monotone sequences

Discontinuity at a domain point: removable, jump and essential cases, and the first- and second-kind grouping

Monotone functions have every relevant finite one-sided limit; interior discontinuities are jumps, with side limits ordered by the direction of monotonicity

Froda's theorem: the set of discontinuities of a monotone function on an interval is at most countable, the injection into ℕ being built from one fixed enumeration of the rationals by least index, so no choice principle is used

Converse to Froda: for every at most countable E ⊆ ℝ there is a bounded nondecreasing f : ℝ → ℝ whose set of discontinuities is exactly E, every one of them a jump

A continuous injective function on an interval is strictly monotone

Continuous inverse theorem: a continuous injective f on an interval I is a bijection onto the order-convex set f[I], and the inverse g : f[I] → I is continuous and strictly monotone in the same sense as f

The Cantor function is continuous on [0,1]

The Cantor function is continuous and nondecreasing, climbs from 0 to 1, and is constant on every interval removed in the construction of the Cantor set, so all of its increase happens on a set of measure zero

The oscillation ω_f(S) = sup{ |f(x) - f(y)| : x, y ∈ S } of f on a set and the oscillation ω_f(c) = inf_δ > 0 ω_f(A ∩ N_δ(c)) at a point, both taken in the extended reals

f : A → ℝ is continuous at c ∈ A if and only if ω_f(c) = 0

For every real ε > 0 the set { x ∈ A : ω_f(x) ≥ ε } is the intersection with A of a closed subset of ℝ; in particular it is closed in ℝ when A = ℝ

For f : A → ℝ the set of points of A at which f is discontinuous is the intersection with A of an F_σ subset of ℝ, and the set of points at which f is continuous is the intersection with A of a G_δ subset; for A = ℝ the two sets are F_σ and G_δ outright

The Dirichlet function 1_ℚ, and Thomae's function t with t(x) = 1/q at a rational x = p/q in lowest terms with q ≥ 1 and t(x) = 0 at every irrational x

The Dirichlet function is continuous at no point of ℝ, and Thomae's function is continuous at every irrational and at no rational, so its set of continuity points is exactly the set of irrationals and its oscillation at c equals t(c)

Every G_δ subset of ℝ is the set of continuity points of some f : ℝ → ℝ, so the G_δ sets are exactly the continuity sets

No function ℝ → ℝ is continuous at every rational and discontinuous at every irrational, because ℚ is not G_δ

The intermediate value property (Darboux property) of a function on an interval: the image of every subinterval is order-convex

Upper and lower semicontinuity of f : A → ℝ at a point of A and on A

f is upper semicontinuous on A if and only if {x ∈ A : f(x) < α} is relatively open in A for every real α, lower semicontinuous if and only if {x ∈ A : f(x) > α} is, and continuous if and only if it is both

Semicontinuous extreme value theorem: an upper semicontinuous function on a nonempty compact K ⊆ ℝ is bounded above and attains a maximum, and a lower semicontinuous one is bounded below and attains a minimum

Baire category inside a closed bounded interval: if [a,b] with a < b is covered by a sequence of closed sets, then one of them contains a nondegenerate closed subinterval of [a,b]; no choice principle is used

Pointwise convergence of a sequence of real functions, and the Baire class one functions as the pointwise limits of sequences of continuous functions

Baire's theorem: a Baire class one function on a closed bounded interval [a,b] is continuous at the points of a dense subset of [a,b] that is the trace of a G_δ set, so its set of discontinuities is meager

Cauchy's functional equation f(x+y) = f(x) + f(y), and the additive functions ℝ → ℝ

An additive f : ℝ → ℝ satisfies f(0) = 0, f(-x) = -f(x) and f(qx) = q f(x) for every rational q and every real x; in particular f(q) = q f(1) at every rational q

If an additive f : ℝ → ℝ is bounded above on some nondegenerate interval, then f(x) = f(1) x for every real x

Six regularity conditions each force an additive f : ℝ → ℝ to be x ↦ f(1)x: continuity at a single point, monotonicity on a nondegenerate interval, boundedness above on one, boundedness below on one, constancy of sign on one, and a graph that is not dense in ℝ²

Assuming the Axiom of Choice, ℝ has a Hamel basis over ℚ: there is B ⊆ ℝ such that every real is a finite ℚ-linear combination of elements of B in exactly one way, and each basis vector carries a well-defined ℚ-linear coefficient map

FALSE: a function with the intermediate value property on an interval is continuous

FALSE: every additive f : ℝ → ℝ is of the form x ↦ cx for a single real c