2. Say (X,d) is a metric space and A is a nonempty subset of X. Define...

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2. Say (X,d) is a metric space and A is a nonempty subset of X. Define a real-valued function f(p) on X by (p) = inf{d(p. x)|2 A}. a) Show that f(p) f(g)| d(p. q) b) Use a) to show that is a continuous function. c) Show that f(p) = 0 if and only if p A. d) Say A is a closed subspace of Euclidean n-space. Show that f(p) = d(p. x) for some r A. Hint: explain why there is a sequence a. a. a3. .... in A with d(p,a,) (p) as j . Does a, have a convergent subsequence? 2. Say (X,d) is a metric space and A is a nonempty subset of X. Define a real-valued function f(p) on X by (p) = inf{d(p. x)|2 A}. a) Show that f(p) f(g)| d(p. q) b) Use a) to show that is a continuous function. c) Show that f(p) = 0 if and only if p A. d) Say A is a closed subspace of Euclidean n-space. Show that f(p) = d(p. x) for some r A. Hint: explain why there is a sequence a. a. a3. .... in A with d(p,a,) (p) as j . Does a, have a convergent subsequence?

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a To show that fp fq dp q we start by noting that for any a A we have dp a 0 and dq a 0 Therefore we ... View the full answer