Continuity, IVT, EVT, and Uniform Continuity

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✓ 17 results · all verified · 11 also independently AI-judged

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1 · Prerequisites

Real Analysis

Set Theory

Topology

2 · Logical flowchart

3 · Items

Definition

Continuity of f : A → ℝ at a point of A and on A: the ε-δ condition, its agreement with lim_x → c f(x) = f(c) at a limit point, and continuity at an isolated point

Theorem

f is continuous at c ∈ A if and only if f(xₖ) → f(c) for every sequence in A converging to c, the converse direction costing countable choice

Theorem

Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function

Theorem

A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs

Theorem

f : A → ℝ is continuous on A if and only if the preimage of every open subset of ℝ is the intersection with A of an open subset of ℝ, and dually for closed sets

Theorem

Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on [a,b] takes every value between f(a) and f(b)

Corollary

The image of an interval under a continuous real function is order-convex, hence an interval, and the image of a closed bounded interval is a closed bounded interval

Theorem

Dictionary: for A ⊆ ℝ with the metric d(x,y) = |x-y|, continuity and uniform continuity of f : A → ℝ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of ℝ is compact in the open-cover sense of ℝ exactly when it is a compact metric subspace

Theorem

Heine-Cantor in ℝ: a continuous real function on a compact subset of ℝ is uniformly continuous

Corollary

A uniformly continuous real function on a subset D ⊆ ℝ extends uniquely to a uniformly continuous function on the closure of D

Theorem

If |f(x) - f(y)| ≤ C|x-y|^α on an interval for some rational α > 1 then f is constant

Theorem

Rudin 4.20, the sharp converse: on a noncompact E ⊆ ℝ there is an unbounded continuous function and a bounded continuous function with no greatest value, and if E is bounded there is a continuous function on E that is not uniformly continuous

False statement

Continuity, IVT, EVT, and Uniform Continuity

3 · Items

Continuity of f : A → ℝ at a point of A and on A: the ε-δ condition, its agreement with lim_x → c f(x) = f(c) at a limit point, and continuity at an isolated point

f is continuous at c ∈ A if and only if f(xₖ) → f(c) for every sequence in A converging to c, the converse direction costing countable choice

Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function

A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs

f : A → ℝ is continuous on A if and only if the preimage of every open subset of ℝ is the intersection with A of an open subset of ℝ, and dually for closed sets

The image of a compact subset of ℝ under a continuous real function is compact

A continuous real function on a compact subset of ℝ is bounded

Extreme value theorem: a continuous real function on a nonempty compact subset of ℝ attains a greatest and a least value

Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on [a,b] takes every value between f(a) and f(b)

The image of an interval under a continuous real function is order-convex, hence an interval, and the image of a closed bounded interval is a closed bounded interval

Every continuous map of a closed bounded interval into itself has a fixed point

A function on an interval satisfying f(x) ≤ f(y) whenever x ≤ y, whose image is order-convex, is continuous

Uniform continuity of f : A → ℝ: one δ serving every pair of points of A

Heine-Cantor in ℝ: a continuous real function on a compact subset of ℝ is uniformly continuous

A uniformly continuous real function on a subset D ⊆ ℝ extends uniquely to a uniformly continuous function on the closure of D

If |f(x) - f(y)| ≤ C|x-y|^α on an interval for some rational α > 1 then f is constant

Rudin 4.20, the sharp converse: on a noncompact E ⊆ ℝ there is an unbounded continuous function and a bounded continuous function with no greatest value, and if E is bounded there is a continuous function on E that is not uniformly continuous

FALSE: every continuous real function is uniformly continuous on its domain

FALSE: the image of a closed subset of ℝ under a continuous real function is closed

FALSE: a continuous real function on a bounded domain attains a greatest value