99年 - 99 國立高雄大學_博士班招生考試試題_應用數學學系：高等微積分#147985

科目：研究所、轉學考(插大)-高等微積分 | 年份：99年 | 選擇題數：0 | 申論題數：8

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所屬科目：研究所、轉學考(插大)-高等微積分

選擇題 (0)

申論題 (8)

1 (10) If $x_n$ and $y_n$ are bounded sequences in $\mathbb{R}$, prove that

$$\limsup(x_n + y_n) \leq \limsup x_n + \limsup y_n.$$
2 (15) A set \(A \subseteq \mathbb{R}^n\) is said to be dense in \(B \subseteq \mathbb{R}^n\) if \(B \subseteq \operatorname{cl}(A)\). If \(A\) is dense in \(\mathbb{R}^n\) and \(U\) is open, prove that \(A \cap U\) is dense in \(U\). Is this true if \(U\) is not open?
3 (20) Suppose that \( f: M \to N \) is continuous and \( B \subseteq M \) is compact. Then

a. Prove that \( f(B) \) is compact.
b. Prove that \( f \) is uniformly continuous on \( B \).
4 (10) Let \( f_n: \mathbb{R} \to \mathbb{R} \) be uniformly continuous and let \( f_n \) converge uniformly to \( f \). Prove that \( f \) is uniformly continuous.
5 (10) Suppose that \( f:[0,1] \to \mathbb{R} \) is continuous, \( f \) is differentiable on \( (0,1) \), and \( f(0) = 0 \). Assume that \( |f'(x)| \leq |f(x)| \), \( 0 < x < 1 \). Prove that \( f(x) = 0 \) for all \( x \in [0,1] \).
6 (15) Let \( M \) be a compact metric space and \( \Phi : M \to M \) be such that \( d(\Phi(x), \Phi(y)) < d(x, y) \) for all \( x, y \in M \), \( x \neq y \). Show that \( \Phi \) has a unique fixed point.
7 Define the function \(f:\mathbb{R}^2\to \mathbb{R}\) by

$$f(x, y) = \begin{cases} \frac{x\sqrt{x^2+y^2}}{|y|} & \text{if } y \neq 0 \\ 0 & \text{if } y = 0 \end{cases}$$

a. (10) Prove that the function \( f \) is not continuous at the point \( (0,0) \).
b. (10) Prove that the function \( f \) has directional derivatives in all directions at the point \( (0, 0) \).