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 to X with respect to the metric dx, the sequence (f(x)) converges to f(z) with respect to the metric dy. 1 (c) Whenever V is an open set in Y, the set f'(V) = {1 € X : f(x) EV) is an open set in X. (d) Whenever F is a closed set in Y, the set f¹(F) = {2 € X : f(x) F} is a closed set in X. 4. Extension to Uniform Continuity: Extend the discussion of this theorem to uniform continuity. How do the statements change if we replace continuity with uniform continuity? Are the equivalent conditions still valid, or do they need modification? Formulate analogous conditions for uniform continuity and discuss their equivalence. 5. Counterexamples Involving Discontinuities: • Provide counterexamples of functions f: XY where X and Y are metric spaces for which one or more of the conditions (a)-(d) fail. For instance, construct a function that satisfies condition (c) but fails condition (a), illustrating why the implications in the theorem do not hold without all metric space requirements. 6. Comparing with Compactness: ■ Analyze how the equivalence of these conditions changes if X is assumed to be compact. Prove that if X is a compact metric space, then a function f : XY is continuous if and only if it satisfies condition (b). Discuss why compactness simplifies the proof and what role it plays in metric space topology.
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 to X with respect to the metric dx, the sequence (f(x)) converges to f(z) with respect to the metric dy. 1 (c) Whenever V is an open set in Y, the set f'(V) = {1 € X : f(x) EV) is an open set in X. (d) Whenever F is a closed set in Y, the set f¹(F) = {2 € X : f(x) F} is a closed set in X. 4. Extension to Uniform Continuity: Extend the discussion of this theorem to uniform continuity. How do the statements change if we replace continuity with uniform continuity? Are the equivalent conditions still valid, or do they need modification? Formulate analogous conditions for uniform continuity and discuss their equivalence. 5. Counterexamples Involving Discontinuities: • Provide counterexamples of functions f: XY where X and Y are metric spaces for which one or more of the conditions (a)-(d) fail. For instance, construct a function that satisfies condition (c) but fails condition (a), illustrating why the implications in the theorem do not hold without all metric space requirements. 6. Comparing with Compactness: ■ Analyze how the equivalence of these conditions changes if X is assumed to be compact. Prove that if X is a compact metric space, then a function f : XY is continuous if and only if it satisfies condition (b). Discuss why compactness simplifies the proof and what role it plays in metric space topology.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.7: Operations On Functions
Problem 63E
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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 to X with respect to the metric dx, the sequence
(f(x)) converges to f(z) with respect to the metric dy.
1
(c) Whenever V is an open set in Y, the set f'(V) = {1 € X :
f(x) EV) is an open set in X.
(d) Whenever F is a closed set in Y, the set f¹(F) = {2 € X :
f(x) F} is a closed set in X.
4. Extension to Uniform Continuity:
Extend the discussion of this theorem to uniform continuity. How do the statements change
if we replace continuity with uniform continuity? Are the equivalent conditions still valid, or
do they need modification? Formulate analogous conditions for uniform continuity and
discuss their equivalence.
5. Counterexamples Involving Discontinuities:
• Provide counterexamples of functions f: XY where X and Y are metric spaces for
which one or more of the conditions (a)-(d) fail. For instance, construct a function that
satisfies condition (c) but fails condition (a), illustrating why the implications in the theorem
do not hold without all metric space requirements.
6. Comparing with Compactness:
■ Analyze how the equivalence of these conditions changes if X is assumed to be compact.
Prove that if X is a compact metric space, then a function f : XY is continuous if and
only if it satisfies condition (b). Discuss why compactness simplifies the proof and what role
it plays in metric space topology.
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