[分析] 實分析
1.
If a,b > 0, let
f(x) = (x^a)*sin(x^-b) for 0 < x <= 1
0 if x = 0
prove that f is of bounded variation in [0,1] if and only if a>b.Then, by
taking a=b, construct (for each 0<α<1) a function that satisfies the
Lipschitz condition of exponent α
|f(x)-f(y)|<= A |x-y|^α
but which is not of bounded variation.
[Hint: Note that if h>0, the difference |f(x+h)-f(x)| can be estimated by
C(x+h)^a, or C'h/x by the mean value theorem. Then ,consider two cases,
whether x^a+1 >h or x^a+1 < h. What is the relationship between α and a
]
2.If F is of bounde variation in [a,b],then:
b
∫|F'(x)|dx = TF(a,b) if and only if F is absolutely continuous.
a
(TF(a,b): total variation)
煩請各位高手幫忙解惑 謝謝
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