p*****n
发帖数: 758 | 1 f是[0,1]上实值连续函数，对任意y \in R，最多有M(与y无关)个点使得f(x)=y.
证明f is of bounded variation. | Q***5
发帖数: 994 | 2 For any partition 0=x_1
that
|y_2-y_1|+|y_3-y_2|+... + |y_n-y_{n-1}|<=M(b-a), where
b=max(f(x)| 0<=x<=1), a = min(f(x)|0<=x<=1)
To prove that, let g_k be the characteristic function on ( min(y_k, y_{k+1})
, max(y_k, y_{k+1}) ) (that is, g_k = 1 on the open segment, and 0 otherwise)
By the condition, g_1+g_2+ ...+ g_{n-1} is positive, but bounded above by M
on [a b], therefore its integral on [a b] is no greater than M(b-a). On the other hand
, t
【在 p*****n 的大作中提到】
: f是[0,1]上实值连续函数，对任意y \in R，最多有M(与y无关)个点使得f(x)=y.
: 证明f is of bounded variation.
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