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MATH 131B Midterm 2 2024
发布时间：2024-06-03
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MATH 131B, Lecture 1
Midterm 2
Fri, May 24, 2024
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.
