匿名使用者
匿名使用者 發問時間: 科學數學 · 1 0 年前

中間值定理

suppose that f: R2→R is continuous and S is a arcwise connected subset ofR2 if a(向量) b(向量)∈S such that f(a)﹤f(b) prove that for any k between f(a) and f(b) there exist c∈S such that f(c)=k

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  • Eric
    Lv 6
    1 0 年前
    最佳解答

    hint: pick an arc

    2007-06-22 06:25:57 補充:

    Proof. Let k be between f(a) and f(b), and let g be an arc connecting a and b; that is, let g:[0,1]→A包含於S be a homeomorphism with g(0) = a and g(1) = b. Then fog (the composition of f and g) is a continuous function on a compact interval with (fog)(0) = f(a) < f(b) = (fog)(1). By the intermediate value theorem, there exists t in [0,1] such that (fog)(t) = k. Set c = g(t), which is in S. Then f(c) = (fog)(t) = k. ::

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