Re: [微積] 高微的積分問題

[Helilo (哈里路)]

※ 引述《Helilo (哈里路)》之銘言：
: Prove the following function is integrable
:          f(x) =   0  ,  x is not a rational number
:                  1/q ,  x = p/q , p,q belong to N, (p,q) = 1
以下根據math1209大大給的提示嘗試作答

為求題目完整先定義 f(0) = 1, 定義 x belongs to [0,1]

Proof:

Because irrational numbers are dense

For any partition P={x_1,x_2,....,x_n}

We can find a irrational number x with x_(i-1) <= x <= x_i for any i

=> m_i = inf{f(x)|x_(i-1) <= x <= x_i} = 0

=> L(P,f)=0

For any ε> 0, exist n > 3/(2ε)

For the fixed n, there are only n(n-1)/2 + 2 of rational numbers p/q

such that 1/q >= 1/n. Let P be a partition of [0,1] such that

Δx_i < 1/n^3 for all i

=> U(P,f) < (n(n-1)/2+2)*1/n^3 + 1/n < 3/(2n) for n large enough

=> U(P,f) < 3/(2n) < ε

=> U(P,f) - L(P,f) < ε

=> f is integrable.

這個證明不知道對不對@@"
而且有點偷懶
要是區間是[a,b]而非[0,1]的話又要重寫了orz

懇請各位賜教<(_ _)>

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