Continuous functions between metric spaces

Content created by Louis Wasserman.

Created on 2025-03-30.
Last modified on 2025-03-30.

module metric-spaces.continuous-functions-metric-spaces where
Imports 
open import elementary-number-theory.positive-rational-numbers

open import foundation.dependent-pair-types
open import foundation.existential-quantification
open import foundation.inhabited-subtypes
open import foundation.propositional-truncations
open import foundation.propositions
open import foundation.subtypes
open import foundation.universe-levels

open import metric-spaces.continuous-functions-premetric-spaces
open import metric-spaces.functions-metric-spaces
open import metric-spaces.metric-spaces

Idea

A function f between metric spaces X and Y is continuous at a point x if there exists a function m : ℚ⁺ → ℚ⁺ such that whenever x' is in an m ε-neighborhood of x, f x' is in an ε-neighborhood of f x. m is called a modulus of continuity of f at x.

Definitions

module _
  {l1 l2 l3 l4 : Level} (X : Metric-Space l1 l2) (Y : Metric-Space l3 l4)
  (f : map-type-Metric-Space X Y)
  where

  is-modulus-of-continuity-at-point-prop-Metric-Space :
    (x : type-Metric-Space X) → (ℚ⁺ → ℚ⁺) → Prop (l1 ⊔ l2 ⊔ l4)
  is-modulus-of-continuity-at-point-prop-Metric-Space =
    is-modulus-of-continuity-at-point-prop-Premetric-Space
      ( premetric-Metric-Space X)
      ( premetric-Metric-Space Y)
      ( f)

  is-modulus-of-continuity-at-point-Metric-Space :
    (x : type-Metric-Space X) → UU (l1 ⊔ l2 ⊔ l4)
  is-modulus-of-continuity-at-point-Metric-Space =
    is-modulus-of-continuity-at-point-Premetric-Space
      ( premetric-Metric-Space X)
      ( premetric-Metric-Space Y)
      ( f)

  is-continuous-at-point-prop-Metric-Space :
    (x : type-Metric-Space X) → Prop (l1 ⊔ l2 ⊔ l4)
  is-continuous-at-point-prop-Metric-Space =
    is-continuous-at-point-prop-Premetric-Space
      ( premetric-Metric-Space X)
      ( premetric-Metric-Space Y)
      ( f)

  is-continuous-at-point-Metric-Space :
    (x : type-Metric-Space X) → UU (l1 ⊔ l2 ⊔ l4)
  is-continuous-at-point-Metric-Space =
    is-continuous-at-point-Premetric-Space
      ( premetric-Metric-Space X)
      ( premetric-Metric-Space Y)
      ( f)

Recent changes