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Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) *Give appropriate graphs and required codes. * Make use of inequalities if you think that required. * You are supposed to use kreszig for reference. Another important characterization of continuous functions involves open sets. Theorem 2.1.5. Let (X,dx) be a metric space, and let (Y,dy) be an- other metric space. Let f : XY be a function. Then the following four statements are equivalent: (a) f is continuous. (b) Whenever (()) is a sequence in X which converges to some point zo X with respect to the metric dx, the sequence (f(z))) converges to f(zu) with respect to the metric dy. (e) Whenever V is an open set in Y, the set f (V): === {x Є X: f(x) EV} is an open set in X. (d) Whenever F is a closed set in Y, the set f¹(F) := f(x) F) is a closed set in X. {x Є X : 1. Proving Equivalence: •Provide a detailed proof showing that conditions (a), (b), (c), and (d) are indeed equivalent. Start with (a) implying (b), then (b) implying (c), and so on, until you complete the cycle back to (a). Discuss the role of open and closed sets in metric spaces and how sequences relate to the topology. 2. Application in Specific Metric Spaces: ⚫ Consider the metric spaces (X, dx) = (R,|-|) and (Y, dy) = (R,|-|), where |-| denotes the usual absolute value metric. Show that a function f: RR is continuous if and only if it satisfies condition (b). Provide examples where you explicitly construct sequences (()) that converge to a point and show whether (f(x("))) converges to f(x0). 3. Behavior Under Different Topologies: ⚫ Let X = R with the usual metric and Y = R with the discrete metric. Analyze how the equivalence of conditions (a) through (d) behaves under these different choices of topologies for Y. Does the theorem still hold in the same way, or are there any exceptions?

Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) *Give appropriate graphs and required codes. * Make use of inequalities if you think that required. * You are supposed to use kreszig for reference. Another important characterization of continuous functions involves open sets. Theorem 2.1.5. Let (X,dx) be a metric space, and let (Y,dy) be an- other metric space. Let f : XY be a function. Then the following four statements are equivalent: (a) f is continuous. (b) Whenever (()) is a sequence in X which converges to some point zo X with respect to the metric dx, the sequence (f(z))) converges to f(zu) with respect to the metric dy. (e) Whenever V is an open set in Y, the set f (V): === {x Є X: f(x) EV} is an open set in X. (d) Whenever F is a closed set in Y, the set f¹(F) := f(x) F) is a closed set in X. {x Є X : 1. Proving Equivalence: •Provide a detailed proof showing that conditions (a), (b), (c), and (d) are indeed equivalent. Start with (a) implying (b), then (b) implying (c), and so on, until you complete the cycle back to (a). Discuss the role of open and closed sets in metric spaces and how sequences relate to the topology. 2. Application in Specific Metric Spaces: ⚫ Consider the metric spaces (X, dx) = (R,|-|) and (Y, dy) = (R,|-|), where |-| denotes the usual absolute value metric. Show that a function f: RR is continuous if and only if it satisfies condition (b). Provide examples where you explicitly construct sequences (()) that converge to a point and show whether (f(x("))) converges to f(x0). 3. Behavior Under Different Topologies: ⚫ Let X = R with the usual metric and Y = R with the discrete metric. Analyze how the equivalence of conditions (a) through (d) behaves under these different choices of topologies for Y. Does the theorem still hold in the same way, or are there any exceptions?

Algebra for College Students
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter8: Functions
Section8.1: Concept Of A Function
Problem 100PS
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Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) *Give appropriate graphs and required codes. * Make use of inequalities if you think that required. * You are supposed to use kreszig for reference. Another important characterization of continuous functions involves open sets. Theorem 2.1.5. Let (X,dx) be a metric space, and let (Y,dy) be an- other metric space. Let f : XY be a function. Then the following four statements are equivalent: (a) f is continuous. (b) Whenever (()) is a sequence in X which converges to some point zo X with respect to the metric dx, the sequence (f(z))) converges to f(zu) with respect to the metric dy. (e) Whenever V is an open set in Y, the set f (V): === {x Є X: f(x) EV} is an open set in X. (d) Whenever F is a closed set in Y, the set f¹(F) := f(x) F) is a closed set in X. {x Є X : 1. Proving Equivalence: •Provide a detailed proof showing that conditions (a), (b), (c), and (d) are indeed equivalent. Start with (a) implying (b), then (b) implying (c), and so on, until you complete the cycle back to (a). Discuss the role of open and closed sets in metric spaces and how sequences relate to the topology. 2. Application in Specific Metric Spaces: ⚫ Consider the metric spaces (X, dx) = (R,|-|) and (Y, dy) = (R,|-|), where |-| denotes the usual absolute value metric. Show that a function f: RR is continuous if and only if it satisfies condition (b). Provide examples where you explicitly construct sequences (()) that converge to a point and show whether (f(x("))) converges to f(x0). 3. Behavior Under Different Topologies: ⚫ Let X = R with the usual metric and Y = R with the discrete metric. Analyze how the equivalence of conditions (a) through (d) behaves under these different choices of topologies for Y. Does the theorem still hold in the same way, or are there any exceptions?
Transcribed Image Text:Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) *Give appropriate graphs and required codes. * Make use of inequalities if you think that required. * You are supposed to use kreszig for reference. Another important characterization of continuous functions involves open sets. Theorem 2.1.5. Let (X,dx) be a metric space, and let (Y,dy) be an- other metric space. Let f : XY be a function. Then the following four statements are equivalent: (a) f is continuous. (b) Whenever (()) is a sequence in X which converges to some point zo X with respect to the metric dx, the sequence (f(z))) converges to f(zu) with respect to the metric dy. (e) Whenever V is an open set in Y, the set f (V): === {x Є X: f(x) EV} is an open set in X. (d) Whenever F is a closed set in Y, the set f¹(F) := f(x) F) is a closed set in X. {x Є X : 1. Proving Equivalence: •Provide a detailed proof showing that conditions (a), (b), (c), and (d) are indeed equivalent. Start with (a) implying (b), then (b) implying (c), and so on, until you complete the cycle back to (a). Discuss the role of open and closed sets in metric spaces and how sequences relate to the topology. 2. Application in Specific Metric Spaces: ⚫ Consider the metric spaces (X, dx) = (R,|-|) and (Y, dy) = (R,|-|), where |-| denotes the usual absolute value metric. Show that a function f: RR is continuous if and only if it satisfies condition (b). Provide examples where you explicitly construct sequences (()) that converge to a point and show whether (f(x("))) converges to f(x0). 3. Behavior Under Different Topologies: ⚫ Let X = R with the usual metric and Y = R with the discrete metric. Analyze how the equivalence of conditions (a) through (d) behaves under these different choices of topologies for Y. Does the theorem still hold in the same way, or are there any exceptions?
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