[助教] 複變小考(王教授班)是非題解答

[blackman1122 (小黑鬼)]

同學們大家好

我是複變(王教授班)的助教

由於蠻多同學來問我小考的是非題的答案

所以在此公布一下是非題的答案

1. The isolated points of a set form a closed set.
   False. A counterexample S = {i/n ; n = 1,2,3...},
          all points are isolated points,
          but zero is a boundary point not in S.

3. The empty set is not an open set, and so it is a closed set.
   False. The empty set is not only an open set, but also a clolsed set.

4. It is impossible to have a neighborhood of x, N(x), that every point in
   N(x) is a boundary point of some set S.
   False. A counterexample; Set S are all rational number in [0,1].
          All points of S are the boundary point of set S.

5. If Cauchy-Riemann equations are satisfied for f (z) = u(x,y)+iv(x,y)
   and u,v ∈ C^1 at a point z, then the function f (z) is analytic at z.
   False. At neighborhood of z.

祝同學考試順利

加油~

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推 b98901097   :推認真助教~ 不過我看不出來(5)為何是False??
推 cktigeryang :是要在z的neiborhood都要符合Cauchy-Riemann eq.
→ cktigeryang :     neighborhood    = =