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MATH 131B continuous function
创建日期：2024-07-10 15:45:29
所属分类：数学mathematics
标签： MATH 131B
continuous function

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MATH 131B, Lecture 1


Midterm 

1. For each of the following conditions, find a continuous function f : (R, dstd ) → (R, dstd ) that satisfies it [and explain why] or show that such a function cannot exist.

(a)  (3 points)  f −1([−1, 1]) is not compact

(b)  (3 points)  f −1(( −∞ , 1)) is compact

(c)  (3 points)  f ([0, 3)) = {−1, 1}

2. Let f : (R, dstd ) → (R, dstd ) be a continuous function that satisfies f (r) = r2  for every r ∈ Q.

(a)  (3 points)  Show that f (√2) = 2

(b)  (3 points)  Write down a function g  :  (R, ddisc )  → (R, dstd )  that  is continuous and satisfies g(r) = r2  for every r ∈ Q but g (√2) ≠ 2. [Be sure to explain why it is continuous!]


3.  (6 points)  Consider the metric space (R, dstd ), the real numbers with the standard metric. For each n, define fn  : R → R by

         Show that the sequence (fn) ∞n=1 converges pointwise to the constant func-tion 0 but NOT uniformly.

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