Continuity, IVT, EVT, and Uniform Continuity: Examples and Counterexamples

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1 · Prerequisites

Real Analysis

Set Theory

Topology

2 · Logical flowchart

3 · Items

Example

The distance ψ(x) = d(x, ℤ) from a real number to the integers is 1-Lipschitz, hence uniformly continuous, takes values in [0,1/2], and vanishes exactly on ℤ

Counterexample

The intermediate value theorem gives a second proof that every nonnegative real has an n-th root, applied to xⁿ on a closed bounded interval

Example

A worked fixed point on [1,2] for the map x ↦ (x + 2/x)/2, from the one-dimensional fixed point theorem

Sources

Continuity, IVT, EVT, and Uniform Continuity: Examples and Counterexamples

3 · Items

The distance ψ(x) = d(x, ℤ) from a real number to the integers is 1-Lipschitz, hence uniformly continuous, takes values in [0,1/2], and vanishes exactly on ℤ

The indicator of ℚ is continuous at no point of ℝ

x · 1_ℚ(x) is continuous at 0 and at no other point

x ↦ 1/x is continuous on (0,1) and not uniformly continuous there, the pairs 1/(k+2) and 1/(k+3) defeating every δ

x ↦ x² is continuous on ℝ and not uniformly continuous, the pairs k+1 and k+1+1/(k+1) defeating every δ

On [0,1] the function x^β is β-Hölder and is α-Hölder for no rational α > β, so the Hölder classes are strictly nested

The identity is uniformly continuous on ℝ and its square is not, so uniform continuity is not preserved by products

The identity on (0,1) is bounded with no greatest value, and on [0,∞) it is continuous and unbounded

The intermediate value theorem gives a second proof that every nonnegative real has an n-th root, applied to xⁿ on a closed bounded interval

A worked fixed point on [1,2] for the map x ↦ (x + 2/x)/2, from the one-dimensional fixed point theorem