- #1
Jamin2112
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Homework Statement
Are you up to the challenge?
Let (X,d) be a metric space and let A be a nonempty subset of X. Define f:X→ℝ by f(x)=inf{d(x,a): a in A}. Prove that f is continuous.
Homework Equations
Definition. Let (X,d) and (Y,p) be metric spaces. A function f:X→Y is continuous at a point a in X provided that for each positive number ε there is a positive number δ such that if x is in X and d(a,x)<δ, then p(f(a),f(x))<ε. A function f:X→Y is continuous provided it is continuous at each point of X.
The Attempt at a Solution
All I've been able to do so far is understand the function f given in the problem. For example, we could have X = {1, 2, 3}, A = {1}, and d(x,a) = |x-a|; then f(1) = 0, f(2) = 1, and f(3) = 2. BUT THIS STEP FUNCTION ISN'T CONTINUOUS! So do I have to assume f is continuous?