[分析] 高微

[suzzdicon (舒玆迪控)]

           n
Let f:A（ R -> R be differentiable with A convex, and suppose

||gard f(x)||≦M for x in A. Prove |f(x)-f(y)|≦M||x-y|| for x,y in A.

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我的解法:

Given any x and y in A

By the mean value theorem, there is a point c between x and y s.t.

f(x)-f(y) ＝ Df(c)(x-y)
          ＝ < grad f(c) , x-y >
          ＝ ||grad f(c)|| ||x-y|| cosθ
          ≦ ||grad f(c)|| ||x-y||
          ≦ M||x-y||

麻煩高手幫我看有沒有推論錯誤

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推 math1209    :ok. 不過...你在第三行那邊兩邊加絕對值.
→ math1209    :還有 c 應該寫落在 x, y 的那條連線上 L(x,y).
→ suzzdicon   :恩 了解 謝謝
→ goodGG      :全部通通加絕對值, |f(x)-f(y)| = |...|
→ suzzdicon   :嗯嗯 忘記了 = =
→ suzzdicon   :謝謝!
→ k6416337    :科溪洗襪子不等式