Prove that A = {x : x € X and ƒÃ(x) = 0}. (iii) Suppose A and B are nonempty disjoint closed subsets of X. Use the function g = fĄ – ƒB to prove that there exist disjoint open sets U and V with ACU and B CV.
Prove that A = {x : x € X and ƒÃ(x) = 0}. (iii) Suppose A and B are nonempty disjoint closed subsets of X. Use the function g = fĄ – ƒB to prove that there exist disjoint open sets U and V with ACU and B CV.
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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Could you teach me how to show (ii), (iii) in detail?
![Let 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 ƒÃ :
fA:
X → R, defined as ƒ₁(x) = d(x, A) satisfies
=
|ƒ^(x) - f^(y)] ≤<d(x, y) Vx, y ≤ X,
and that fд is continuous on X.
(ii) Prove that A = {x: xe X and f₁(x) = 0}.
(iii) Suppose A and B are nonempty disjoint closed subsets of X. Use the
function g = fA - ƒB to prove that there exist disjoint open sets U and V
with ACU and B C V.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2Fa34596c4-d763-441e-8fbc-49ce912130b2%2F3qur5mp_processed.png&w=3840&q=75)
Transcribed Image Text:Let 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 ƒÃ :
fA:
X → R, defined as ƒ₁(x) = d(x, A) satisfies
=
|ƒ^(x) - f^(y)] ≤<d(x, y) Vx, y ≤ X,
and that fд is continuous on X.
(ii) Prove that A = {x: xe X and f₁(x) = 0}.
(iii) Suppose A and B are nonempty disjoint closed subsets of X. Use the
function g = fA - ƒB to prove that there exist disjoint open sets U and V
with ACU and B C V.
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