课程名称：高等微积分(一) 课程性质：必修 课程范围：课本Ch1~Ch3 开课教师：蔡纹琦 开课学院：商学院 开课系级：统计二 考试日期(年月日)：2008/11/18 考试时限(Mins)：二234 试题本文： 1.(40 points) For each of the following statements, determine whether it is true or false . (a)___The set Z of integers is dense in R. (b)___If the sequence {an^2} converges, then the sequence {an} also converges. (c)___Let {an} be a sequence of real numbers. If for some ε﹥0, there is an index N such that |an| ﹤εfor all indices n≧N, then {an} converges to 0. (d)___A subsequence of a convergent sequence is convergent. (e)___Suppose that the sequence {an} is monotone and that it has a convergent subsequent subsequence. Then {an} converges. (f)___The set [0,∞) is closed. (g)___Every continuous function f:(0,1)→R has a bounded image. (h)___Suppose that the function f:[0,1]→R is continuous and its image is con- tained in set of rational numbers. Then f is a constant function. (i)___Any Lipschitz function is uniformly continuous.(A function f:D→R is sa- id to be Lipschitz function provided that there is a nonnegative number C such that |f(u)-f(v)|≦C|u-v| for all u and v in D. (j)___If f:[0,1]→R is continuous and one-to-one, then f is strictly monotone. 2.(6 points) Please state the Completeness Axiom which the set of real number satisfies. 3.(6 points) Please state the Intermediate Value Theorem. 4.(10 points) Let A and B be compact sets. Show that the union A∪B is also compact. 5.(10 points) Suppose that the function f:R→R is continuous at the point x0 and f(x0)＞0. Then f(x) is positive around x0. Namely, please show that there is an interval I = (x0 -1/N, x0 +1/N), where N is a natural number, such that f(x)＞0 for all x in I. 6.(8 points) Define f(x) = x if x is rational, = -x if x is irrational. Please locate the continuous points and discontinuous points. Justify your answer. 7.(10 points) Give an example of a function defined on (0,1) such that f is continuous and bounded but not uniformly continuous in (0,1). Justify your answer. 8.(10 points) Given a function :f:[-1,1]→R, define functions g and h on [-1,0)∪(0,1] by g(x)≡f(x)/x and h(x)≡f(x)/(x^2) . Please find an example of f having the property that there is some M＞0 such that |f(x)|≦M|x|^2 for all x in [-1,1], and lim g(x) exists but lim h(x) doesn't exist. Justify your x→0 x→0 answer. --

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