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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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