Eritque Arcus Math Notes

uniformly continue

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Def pg. 143

let 𝐴⊆ℝ and let 𝑓:𝐴⟶ℝ. We say 𝑓 is uniformly continue if
(∀𝜀>0)(∃𝛿(𝜀)>0)(∀𝑥,𝑢∈𝐴)𝑑(𝑥,𝑢)<𝛿(𝜀)⟹𝑑(𝑓(𝑥),𝑓(𝑢))<𝜀

Thm Uniformly continuity theorem pg. 143

Let 𝐼 be a bounded closed interval and let 𝑓:𝐼⟶ℝ be continue on 𝐼. Then 𝑓 is uniformly continue on 𝐼.

Thm pg.144

If 𝑓:𝐴⟶ℝ is uniformly continue on 𝐴⊆ℝ and if (𝑥𝑛) is Cauchy sequence in 𝐴, then (𝑓(𝑥𝑛)) is Cauchy sequence in ℝ

Thm 27.1

For 𝑓:𝐶⟶𝑦 is continue and 𝐶 is compact, we can say 𝑓 is uniformly continue.

Thm 27.2

For 𝐶 compact, 𝑦 complete, 𝑓:𝐷⟶𝑦 and 𝐷⊂𝐶 is dense, TFAE:

  1. 𝐹:𝐶⟶𝑦 continue and 𝐹∣𝐷=𝑓(means 𝑓 is 𝐹 when domain is restricted to 𝐷)
  2. 𝑓 is uniformly continue

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