※ 引述《zidanesquall (請前面大大們高抬貴手)》之銘言：
: 1.Show that both the sum and product of absolutely continous functions are
: absolutely continous functions are absolutely continuouse.
: 2.Define f on R by
: xsin(1/x) if x≠0,
: f(x)={
: 0 if x=0
: Find the upper and lower derivatives of f at x=0.
: 想請問一下版上的前輩，這兩題要怎麼做，我的基礎還不太夠，想了很久還是想不出來..
1. f, g absolutely continuous. 對所有ε> 0, 存在δ(ε)使f, g滿足“那個條件”
任意給定一個ε, 則δ(ε/2)會使f + g滿足“那個條件”, 根據三角不等式:
|f(x) + g(x) - f(y) - g(y)| <= |f(x) - f(y)| + |g(x) - g(y)|
設|f|, |g| < M, 則δ(ε/2M)會使fg滿足“那個條件”, 根據這個不等式:
|f(x)g(x) - f(y)g(y)| <= |f(x) - f(y)| |g(x)| + |g(x) - g(y)| |f(y)|
2. 視覺上來看, upper = 1, lower = -1, x挑 1/π, 1/3π, ... 之類的來驗就好了
另一方面, 顯然f(x)會被 g(x) = x, h(x) = -x二函數給夾住.
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