math104-s21:s:kelvinlee
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math104-s21:s:kelvinlee [2021/05/11 10:39] 202.81.230.88 [Questions] |
math104-s21:s:kelvinlee [2026/02/21 14:41] (current) |
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| ===== Questions ===== | ===== Questions ===== | ||
| - | 1. What's the difference between ? \\ | + | 1. What's the difference between |
| 2. What's the difference between pointwise convergence and uniform convergence? | 2. What's the difference between pointwise convergence and uniform convergence? | ||
| 3. Is a series of continuous functions necessarily continuous? \\ | 3. Is a series of continuous functions necessarily continuous? \\ | ||
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| \lim _{n \rightarrow \infty} \int_{a}^{b} f(x) \sin (n x) d x=0. | \lim _{n \rightarrow \infty} \int_{a}^{b} f(x) \sin (n x) d x=0. | ||
| $$ | $$ | ||
| - | 5. (Yuwei Fan's practice final) Suppose that $f:[1, \infty) \rightarrow \mathbb{R}$ is ]$. Prove that there exists $M>0$ such that $\frac{|f(x)|}{x} \leq M$ holds for any $x \geq 1$. \\ | + | 5. (Yuwei Fan's practice final) Suppose that $f:[1, \infty)$ $\rightarrow \mathbb{R}$ is \text{uniformly continuous} on $[1, \infty)$. Prove that there exists $M>0$ such that $\frac{|f(x)|}{x} \leq M$ holds for any $x \geq 1$. \\ |
| 6. What are some nice properties that continuity preserves? \\ | 6. What are some nice properties that continuity preserves? \\ | ||
| ** Answer: ** Compactness, | ** Answer: ** Compactness, | ||
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| 37. A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to satisfy the Lipshitz condition of order $\alpha$ at $a \in \mathbb{R}$ if there is a constant $M$ and a neighborhood of $a$ such that | 37. A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to satisfy the Lipshitz condition of order $\alpha$ at $a \in \mathbb{R}$ if there is a constant $M$ and a neighborhood of $a$ such that | ||
| $$ | $$ | ||
| - | |f(x)-f(a)|< | + | \left|f(x)-f(a)\right|< |
| $$ | $$ | ||
| Show that if $f$ has the Lipschitz condition of order $\alpha$ for $\alpha> | Show that if $f$ has the Lipschitz condition of order $\alpha$ for $\alpha> | ||
math104-s21/s/kelvinlee.1620729599.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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