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2. Part (c) of Theorem 2.7.25 in the lecture notes told us that if f: R" → Rm is continuous, then for any closed set CCRm, f-1(C) is closed. With this in mind, for each of the following statements, either prove the statement or provide a counterexample. When providing a counterexample, you can choose your m and n. If you are proving a statement, you should do this in full generality; arbitrary m and n.: (a) If ƒ : Rn → Rm is a function so that f(C) is closed for any closed set CCR", then f is continuous. (b) If ƒ : R¹ → Rm is a function so that f(C) is closed for any closed set CCR, then f(0) is open for any open set OCRn. (c) If f: R → Rm is continuous, then f(C) is closed for any closed set CCR.

2. Part (c) of Theorem 2.7.25 in the lecture notes told us that if f: R" → Rm is continuous, then for any closed set CCRm, f-1(C) is closed. With this in mind, for each of the following statements, either prove the statement or provide a counterexample. When providing a counterexample, you can choose your m and n. If you are proving a statement, you should do this in full generality; arbitrary m and n.: (a) If ƒ : Rn → Rm is a function so that f(C) is closed for any closed set CCR", then f is continuous. (b) If ƒ : R¹ → Rm is a function so that f(C) is closed for any closed set CCR, then f(0) is open for any open set OCRn. (c) If f: R → Rm is continuous, then f(C) is closed for any closed set CCR.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
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2. Part (c) of Theorem 2.7.25 in the lecture notes told us that if f : R" → Rm is continuous, then for any closed set CCRm, f−¹(C) is closed. With this in mind, for each of the following statements, either prove the statement or provide a counterexample. When providing a counterexample, you can choose your m and n. If you are proving a statement, you should do this in full generality; arbitrary m and n.: (a) If f: R" → Rm is a function so that f(C) is closed for any closed set CCR, then f is continuous. (b) If ƒ : R¹ → Rm is a function so that f(C) is closed for any closed set CCR, then f(0) is open for any open set OCRn. (c) If f: R → Rm is continuous, then f(C) is closed for any closed set CCR".
Transcribed Image Text:2. Part (c) of Theorem 2.7.25 in the lecture notes told us that if f : R" → Rm is continuous, then for any closed set CCRm, f−¹(C) is closed. With this in mind, for each of the following statements, either prove the statement or provide a counterexample. When providing a counterexample, you can choose your m and n. If you are proving a statement, you should do this in full generality; arbitrary m and n.: (a) If f: R" → Rm is a function so that f(C) is closed for any closed set CCR, then f is continuous. (b) If ƒ : R¹ → Rm is a function so that f(C) is closed for any closed set CCR, then f(0) is open for any open set OCRn. (c) If f: R → Rm is continuous, then f(C) is closed for any closed set CCR".
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