課程名稱︰分析導論優二 課程性質︰數學系大二必修 課程教師︰王振男 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2015/05/12 考試時限（分鐘）：180 試題 : 1. Evaluate the following integrals. (a) (10%) ∬ [x+y] dA, where Q = [0, 2] ×[0, 2], and [t] is the greatest Q integer ≦ t. (b) (10%) ∬ f(x, y) dA, where Q = [0, 1] ×[0, 1] and Q ╭ 0, if at least one of x, y is irrational, f(x, y) = ╯ ╰ 1/n, if y is rational and x = m/n, where m and n are relatively prime integers with n > 0. 2. (15%) Give an example to show that φ(f(x)) may not be measurable if φ and f are measurable. n 3. (20%) Let f be an extended real-valued measurable function defined on |R . Show that there exists a Borel function g such that f = g a.e. Hint: First consider f ≧ 0 and use the approximation of simple functions. For the + - general case, we write f = f - f . 4. (25%) Show that there exists a measurable set which is not Borel. You could use the fact that a properly-defined inverse of Cantor function, P: [0, 1] → [0, 1], is measurable, 1-1, and the values of P lie entirely in the Cantor set. Also, you may need the fact that if f is a real measurable function on -1 |R and B is a Borel set, then f (B) is a measurable set. 1 1 5. (20%) This is a simple form of Sard's theorem. Let f: I → |R be a C 1 function, where I is an open set of |R . Denote C = {x∈I: f'(x) = 0}. Show that f(C) is of measure zero. Hint: Let K be a closed interval contained in I. consider the function g: K ×K → |R ╭ f(y) - f(x) - f'(x)(y-x) │ ────────────, for x ≠ y, g(x, y) = ╯ y - x │ ╰ 0, for x ＝ y. Then show that g is uniformly continuous. Thus for any ε > 0 there exists δ > 0 such that |g(x, y)| < ε whenever |x-y| < δ. partition K into subintervals with norm less then δ. Then consider those subintervals that have nonempty intersection with C. 註: 第四題原題目敘述並不清晰，這裡的題目敘述是修改後的。 -- 2 2 1 ψxavier13540 給定一個二次元(|R )上的開集 G，設 f: G →|R ∈ C 。考慮一 autonomous system ╭dx/dt = f(x)，若 ∀t ≧ 0，有φ (x°) ∈ K ⊆ G，其中 K 在 G 上 compact，則 ╰x(0) = x° t ω(x°) 只能是一定點、一週期軌道或連接有限個 critical point 的連通路徑，不會像三 次元一樣可能出現混沌(chaos)。此即為 ODE 動力系統中的 Poincaré–Bendixson 定理。 --

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