## 本次英国代写主要为真实分析相关的限时测试

2. 有界函数：例子

f(x) =p1 + x2 在 (0; 1) 上是无界的
f(x) =p1 + x2 在 (1;1) X 上是无界的
g(x) =p1 + (1=x2) 有界于 (0; 1)
g(x) =p1 + (1=x2) 在 (1;1) 上是无界的
h(x) = xp1 + (1=x2) 有界于 (1;1)其他都没有
3. 连续性：定义

8a > 0 9b > 0 使得如果 jx x j < b，则 jf(x) f(x)j < a
X
8a > 0 9b > 0 使得如果 jx x j < a，则 jf(x) f(x)j < b
9a > 0 8b > 0 使得如果 jx xj < a，则 jf(x ) f(x)j < b
9a > 0 8b > 0 使得如果 jx xj < b，则 jf(x ) f(x)j < a
8a > 0 9b > 0 使得如果 jf(x) f(x)j < a，则 jx x j < b

4. 连续性和代数运算：理论

h1 仅 X

h1、h2 和 h3

5. 一致连续性：理论

(i) f 在 ( 2; 2) 上一致连续
(ii) f 在 ( 1; 1) 上一致连续
(iii) f 有界于 ( 2; 2)
(iv) f 有界于 ( 1; 1)

(i) 仅
(ii) 仅
(iii) 仅
(i) 和 (ii) 仅
(ii) 和 (iv) 只有 X

2. Bounded functions: examples
Which of the following statements is true?
 f(x) =
p1 + x2 is unbounded on (0; 1)
 f(x) =
p1 + x2 is unbounded on (1;1) X
 g(x) =
p1 + (1=x2) is bounded on (0; 1)
 g(x) =
p1 + (1=x2) is unbounded on (1;1)
 h(x) = x
p1 + (1=x2) is bounded on (1;1)
 None of the others
3. Continuity: de nitions
A function f is called continuous at the point x 2 R, if . . .
 8a > 0 9b > 0 such that if jx xj < b, then jf(x) f(x)j < a
X
 8a > 0 9b > 0 such that if jx xj < a, then jf(x) f(x)j < b
 9a > 0 8b > 0 such that if jx xj < a, then jf(x) f(x)j < b
 9a > 0 8b > 0 such that if jx xj < b, then jf(x) f(x)j < a
 8a > 0 9b > 0 such that if jf(x) f(x)j < a, then jx xj < b
 None of the others
4. Continuity and algebraic operations: theory
Let f and g be functions on R. Suppose that f is continuous at zero
and g has a jump discontinuity at zero. Let h1 = f + g, h2 = fg and
h3 = f  g. Which functions must have a discontinuity at zero?
 h1 only X
 h2 only
 h3 only
 h1 and h2 only

 h1, h2 and h3
 None of the others
5. Uniform continuity: theory
Let f be a continuous function on the interval ( 2; 2). Consider the
following statements:
(i) f is uniformly continuous on ( 2; 2)
(ii) f is uniformly continuous on ( 1; 1)
(iii) f is bounded on ( 2; 2)
(iv) f is bounded on ( 1; 1)
Which statements must be true for f?
 (i) only
 (ii) only
 (iii) only
 (i) and (ii) only
 (ii) and (iv) only X
 None of the others

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