面白い数学の問題おしえて～な 44問目 (230ﾚｽ)
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164: １３２人目の素数さん [sage] 2025/07/25(金) 15:33:49.02 ID:rZ8dIIWn(1/2) AAS
Fix n∈ℕ. 𝒪ₙ := { I ; I is open interval st. diam( f(I) ) < 1/n }. As f is right continuous, we can find q(x)>x such that (x,a(x)) ∈𝒪ₙ q(x)∈ℚ

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165(1): １３２人目の素数さん [sage] 2025/07/25(金) 15:33:58.76 ID:rZ8dIIWn(2/2) AAS
Suppose there exists x,y ∈ ℝ such that q(x) = q(y). If x < y, then y∈ (x,q(x)) ⊂ ∪𝒪ₙ. Thus y cannot be in ℝ＼∪𝒪ₙ. Thus at least one of x,y cannot be in ∪𝒪ₙ. Thus the restriction q on ℝ＼∪𝒪ₙ is injective. Thus ℝ＼∪𝒪ₙ is countable. Thus ℝ＼∩ₙ ∪𝒪ₙ is also countable. On the other hand f is continuous on ℝ＼∩ₙ ∪𝒪ₙ.

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