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

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?

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 \( x \in [0,1] \).

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 \( (0, 0) \).