EDB — 1HR

[1HR] Let \((X_ 1,d_ 1)\) and \((X_ 2,d_ 2)\) be metric spaces. Let \(\mathcal F\) be a family of functions \(f:X_ 1→ X_ 2\), we will say that it is an equicontinuous family if one of these equivalent properties holds.

• \(∀ \varepsilon {\gt}0\) \(∃𝛿{\gt}0\) \(∀ f∈\mathcal F\)

  \[ ∀ x,y∈ X_ 1,~ d_ 1(x,y)≤ 𝛿⇒ d_ 2(f(x),f(y))≤ \varepsilon \quad . \]

• There exists a a fixed monotonically weakly increasing function \(𝜔:[0,∞)→ [0,∞]\), for which \(\lim _{t→ 0+}𝜔(t)=𝜔(0)=0\) (\(𝜔\) is called ”continuity modulus” ¹ ) such that

  \begin{equation} ∀ f∈{\mathcal F}, ~ ∀ x,y∈ X_ 1, ~ d_ 2(f(x),f(y))≤ 𝜔\big(d_ 1(x,y)\big)\quad . \label{eq:equicontinua} \end{equation}

  4

• There exists a fixed continuous function \(𝜔:[0,∞)→ [0,∞]\) with \(𝜔(0)=0\) such that 4 holds.

(The result [150]↺↻ can be useful to prove equivalence of the last two clauses.)