推 math1209:很溫馨的題目 = = 02/25 18:40
→ clouddeep:大師為什麼這麼說? XD 02/25 18:47
1.Let f:(-1,2)→R be a real analytic function.
If f(1/k)=0 for all natural k, show that f is identically zero.
2.Let f be a nonnegative real-valued function defined on [0,1].
Suppose that there is an universal constant M>0 such that
f(x_1)+....+f(x_n)≦M for every finite subset {x_1,...,x_n} of [0,1].
Show that the set S={x|f(x)=/=0} is countable.
3.Letf:[0,1] → R be a continuous function.
Consider the sequence of functions
x
f_0 = f, f_(n+1)(x) = ∫ f_n(t)dt ,n=0,1,2,...., x in [0,1].
0
Show that f_0(x) + f_1(x) + f_2(x) +...... converges uniformly.
4.A real valued function f(x) is convex function if
f(tc+(1-t)d)≦tf(c)+(1-t)f(d) for all a<c<d<b and 0≦t≦1.
Prove that f is differential convex function on (a,b) iff f'(x) is
increasing on (a,b).
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