Eritque Arcus Math Notes

continue

1 min read

Def 16.1

𝑓 is continue imply
𝑑(𝑥,𝑥′)<𝜀⟹𝑑(𝑓(𝑥),𝑓(𝑥′))<𝜀

Thm 16.2

𝑓 is continue at x ⟺ (lim𝑛𝑥𝑛=𝑥⟹lim𝑛𝑓(𝑥𝑛)=𝑓(𝑥))

Thm 16.3 17.2

𝑓:𝑥⟶𝑦 TFAE

  1. 𝑓 is continuous
  2. it’s inverse image 𝑓−1(𝑂) is open for all open image 𝑂

Thm 17.5

A Metric Space (𝑋,𝑑) and 𝑓,𝑔∈𝐶(𝑋,ℝ)

  1. 𝑓+𝑔∈𝐶(𝑋,ℝ)
  2. 𝑓×𝑔∈𝐶(𝑋,ℝ)
  3. ∃𝜙:ℝ⟶ℝ c.t. ⟹𝜙⚬𝑓∈𝐶(𝑋,ℝ)
    where 𝐶 is the set of all continue functions from 𝑋 to ℝ

Thm 17.7

The uniform limit of continuous functions is continue

Thm

continue ⟺ sequentially continue

Thm 35.2

inverse function of continue function is continue.

Graph View

Backlinks