證明: f:[0,1]-->[0,1] is continuous, then prove there exists a c∈[0,1] such that f(c)=c 用中間值定理證明 拜託各位了~~?

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  • 匿名使用者
    4 年前
    最佳解答

    【定理】

    中間值定理 (Intermediate value theorem) 指出對於連續函數 f : [a, b] → ℝ,若 f(a) < u < f(b) 或 f(a) > u > f(b),則存在 c ∈ (a, b) 使得 f(c) = u。

    【題目】

    f : [0, 1] → [0, 1] is continuous, then prove there exists a c ∈ [0, 1] such that f(c) = c.

    【證明】

    Let g(x) = f(x) - x, then g is continuous on [0, 1].

    Consider g(0) = f(0) - 0 = f(0) ≥ 0.

    Case 1: g(0) = 0

    If g(0) = 0, then f(0) = 0 and c = 0, then the proof is done.

    Consider g(1) = f(1) - 1 ≤ 1 - 1 = 0.

    Case 2: g(1) = 0

    If g(1) = 0, then f(1) = 1 and c = 1, then the proof is done.

    Case 3: g(0) ≠ 0 and g(1) ≠ 0

    If g(0) ≠ 0 and g(1) ≠ 0, then g(0) > 0 and g(1) < 0.

    That is, g(0) > 0 > g(1).

    By intermediate value theorem, there exists c ∈ (0, 1) such that g(c) = 0,

    that is, f(c) - c = 0,

    that is, f(c) = c.

    Therefore, combining 3 cases, there exists a c ∈ [0, 1] such that f(c) = c.

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