→ q0300768:感謝^^ 02/23 22:40
→ clouddeep:我不知道有沒有錯喔....你要自己再仔細想想@@||| sorry 02/23 22:58
※ 引述《q0300768 (NANA真好看~^^)》之銘言：
: 交92(V)
: (a) Let a and b be real numbers with a<b, suppose that the function
: f:(a,b) → R is bounded and is monotonic increasing. Prove that
: both lim f(x) and lim f(x) exist, and so f can be extended to a
: x→a+ x→b-
: continuous function on [a,b].
f:(a,b) → R is bounded and is monotonic increasing
=>f attain its supremum and infimum on (a,b)
=>Assume that M and m are the supremum and infimum of f, respectively.
Show that lim f(x) = M and lim f(x) = m.
x→b- x→a+
If not, then |f(b-1/n_k)-M| > k for all natural k and for some real k>0.
But it shows that f is unbounded on (a,b), so that lim f(x) = M.
x→b-
Similarly, lim f(x) = m.
x→a+
For any x in (a,b), let (x_n) be an arbitrarily sequence converges to x,
so that f(x_n) converges to f(x).(這邊應該可以用monotone的性質來証，給你試看看)
: 清94
: 1. Let f be a continuous real-valued function defined on [a,b], and
: a n 1/n
: let M=max∣f(x)∣.Show that lim (∫∣f(x)∣dx) = M
: xε[a,b] n→∞ b
: 清93
: 5. Let f be real-valued, differentiable function on R such that
: f'(x)>f(x) for all xεR. Assume that f(0)=0, show that f(x)>0
: for all x>0.
大略的寫一下：
f'(0)>f(0) => f'(0)>0.
Applying Mean Value Theorem:
f(x) = f(x)-f(0) = f'(c)(x-0) = xf'(c), note that c is dependent on x.
If f'(c)<0, f(x)<0.
改寫c為x_1.
f(x) = xf'(x_1) > xf(x_1) > x*x_1*f(x_2) >....>x*x_1*...x_n*f(x_n)>....
until x_k is closed enough to 0 for sufficiently large k,
f'(x_k)>0 since f'(x_k)>f(x_k)>0
by the continuity of f on the neighborhood of 0.
Hence f(x)>0 for all x>0.
: 想很久都沒啥頭緒,麻煩板上高手了
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有誤請指正...頭昏了
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