[分析] 絕對連續

[znmkhxrw (QQ)]

請問一下下面這件事會對嗎？

if f：[a,b]→R , g：[c,d]→R , f,g are absolutely continuous , g([c,d])<=[a,b]
                                                                     (被包含)

then f(g(x))：[c,d]→R is absolutely continuous

我照定義做得時候：

for all e>0, there exists common d_e>0 s.t.

1.Σ│f(b_i)-f(a_i)│<e whenever Σ(b_i-a_i)<d_e and [a_i,b_i] nonoverlapping

2.Σ│g(d_i)-g(c_i)│<e whenever Σ(d_i-c_i)<d_e and [c_i,d_i] nonoverlapping

for 2. , take e = d_e , then 2. becomes：

Σ│g(d_i)-g(c_i)│<d_e when Σ(d_i-c_i)<d_(d_e) and [c_i,d_i] nonoverlapping


so take D_e=d_(d_e) , when Σ(d_i-c_i)<D_e and [c_i,d_i] nonoverlapping

we have Σ│f(g(d_i))-f(g(c_i))│ < e by 1. ????????????

目前無法辦到，因為雖然在Σ(d_i-c_i)<D_e下，我們有Σ│g(d_i)-g(c_i)│<e了

可是無法確定每個 [g(d_i),g(c_i)] 是 nonoverlapping，所以不能套入1.

有技巧可解決嗎？還是真的有反例？

謝謝



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