내 부족한 영어 실력이지만 굳이 영어로 올려보겠다... 짤리지만 않았음 좋겠음


f(a)<k<f(b),

E:={x｜x∈[a,b], f(x)≤k} (b is one of the upper bound on E)

Thus, ∃c=lub{E} ∈ ℝ (or sup E)

(f is continuous, so δ＞0 is exist) 


if f(x)<k,  ∀a≤x＜a+δ


if f(x)>k, ∀b-δ<x≤b


∴ a<c<b, Now we want to show f(c)=k, so  we need to show f(c)>k and f(c)<k is not correct.


First, f(c)>k, f(c)-k>0

∃η>0 (∵ f is continuous)


|x-c|<η ⇒ |f(x)-f(c)|<f(c)-k, at the open interval (c-η, c+η), when f(x)>k, 'c-η' can be upper bound on E, but c was l.u.b, so It is contradiction!


Second, f(c)<k, k-f(c)>0

∃ε>0 (∵ f is continuous)


|x-c|<ε ⇒ |f(x)-f(c)|<k-f(c), also can be upper bound, so It is contradiction too. ∴ f(c)=k


하아... 디시인사이드 게시판 기호 좀 추가 좀 해주세요 ㅠㅠㅠㅠ