et A be a non-empty subset of a metric space (X,d) and x an element of X. Define the distance from x to A as d(x, A) = inf{d(x, a) : a ¤ A}. (i) Prove that the function fĄ: X → R, defined as ƒÃ(x) = d(x, A) satisfies |ƒÃ(x) — ƒ^(y)| ≤ d(x, y) Vx, y ≤ X, and that fa is continuous on X. (ii) Prove that A = {x : x € X and f₁(x) = 0}. iii) Suppose A and B are nonempty disjoint closed subsets of X. Use the function g = fA- fB to prove that there exist disjoint open sets U and V with ACU and BCV.

et A be a non-empty subset of a metric space (X,d) and x an element of X. Define the distance from x to A as d(x, A) = inf{d(x, a) : a ¤ A}. (i) Prove that the function fĄ: X → R, defined as ƒÃ(x) = d(x, A) satisfies |ƒÃ(x) — ƒ^(y)| ≤ d(x, y) Vx, y ≤ X, and that fa is continuous on X. (ii) Prove that A = {x : x € X and f₁(x) = 0}. iii) Suppose A and B are nonempty disjoint closed subsets of X. Use the function g = fA- fB to prove that there exist disjoint open sets U and V with ACU and BCV.

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.3: Rates Of Change And Behavior Of Graphs

Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...

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