→ math1209:ex 7.28 (p. 308～). 當然，如果你學過測度論那這會變得 02/24 06:40
→ math1209:很單純．．． 02/24 06:40
推 Potervens:the set of discontinuities has measure zero 02/24 10:59
※ 引述《LuisSantos ( )》之銘言：
: 1.
: x dy - y dx
: Let ω = -------------- , (x , y) 屬於 R^2 - {0} .
: x^2 + y^2
: Show that ω is a closed 1-form , but not exact on R^2 - {0} .
這個我不熟，但我猜只要套 Rudin 第十章的樣子就可以搞定的樣子。
: 2. Let S = {x屬於R^n | ||x|| ≦ 1} and f : S→R be a nonnegative continuous
: function.
: (a) Prove that f has the absolute maximum value on S.
: (b) Let M be the absolute maximum value of f on S .
: k 1/k
: Show that lim (∫(f(x)) dx_1 dx_2 ... dx_n) = M .
: k→∞ S
這一題跟某一題很像：
可參見：http://frankmath.cc/plover/Apostol.pdf (p.302 之後).
你自己依樣畫葫蘆就可以了。
: 3. Prove or disprove (by a counter example) the following statements:
: If f : R→R is a continuous function , then f is an open mapping .
這不對! 因為定理是這個：
Prove that every continuous open mapping of |R into |R is monotonic.
這也是 Rudin 的習題。
請至 bbs://bs2.to 版名:P_Apostol,
精華區-高等微積分-2 ◆ 高微書簡介－個人看法-6 ◆ [Rudin] 高微解答
Ch4. ex 15.
: 4. Let f : [0,1]→R be defined by
: 0 if x is irrational
: f(x) =
: 1 p
: --- if x = ---
: q q
: where p , q ≧ 0 with no common factor .
: Is f integrable on [0,1] ？
這就是鼎鼎大名的 爆米花函數 (或稱為 Riemamnn function). 跟上一題一樣，
Ch4, ex. 18.
: 5. Let C([0,1]) be the set of real continuous functions on [0,1] .
: Show that the complement of the following set
: 1
: A = { f屬於 C([0,1]) | 0 < ∫ f(x) dx < 3 } is closed
: 0
這個我想應該也只是定義的使用。不行我們再討論討論。
: 6. Let (M,d) be a metric space . If A is compact in M and B is closed in M ,
: and A∩B = ψ . Show that there is an δ > 0 such that d(x,y) > δ for
: all x 屬於 A and y 屬於 B .
Rudin 習題，Ch4. ex. 21.
: 7. Let k(x,y) be a continuous real-valued function on the square
: S = [0,1] ╳ [0,1] . Assume that |k(x,y)| < 1 for each (x,y) 屬於 S .
: Let A : [0,1]→R be continuous . Prove that there is a unique continuous
: real-valued function f(x) on [0,1] such that
: 1
: f(x) = A(x) + ∫ k(x,y)f(y) dy by contraction mapping theorem .
: 0
這個只是用他給的 hint: Contraction Mapping Theorem :-)
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