# sa3 - ρ E the distance function from a point x ∈ X to E...

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Self-Assessment Questions – Math 118A, Fall 2009 1. Every rational number can be written in the form x = m/n , where n > 0, and m and n are integers without any comon divisors. When x = 0, we take n = 1. Consider the functions f deFned on R by f ( x ) = ± 0 if x R \ Q , 1 n if x = m n . Prove that f is continuous at every irrational point, and that f has a simple discontinuity at every rational point. 2. Suppose f is a real function with domain R which has the intermediate value property: If f ( a ) < c < f ( b ), then f ( x ) = c for some x between a and b . Suppose also, for every rational r , that the set { x R | f ( x ) = r } is closed. Prove that f is continuous. 3. If E is a nonempty subset of a metric space ( X, d ), we denote by
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Unformatted text preview: ρ E the distance function from a point x ∈ X to E : ρ E ( x ) = inf y ∈ E d ( x, y ) . Prove that ρ E is uniformly continuous. 4. Let ( X, d X ) and ( Y, d Y ) be metric spaces, and f : X → Y a continuous function. If X is compact, prove that f is uniformly continuous. 5. Is it true that if f : R k → R k is continuous and A ⊂ R k is convex, then f ( A ) is convex? 6. Consider a normed space ( X, k · k ). Prove that the closure of an open ball is the closed ball, i.e., { x ∈ X |k x k < R } = { x ∈ X |k x k ≤ R } . 1...
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