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4. Consider the set of all continuous functions f: R → R: 8(R) = {ƒ :R→R | ƒ is a continuous function}, This set forms a ring under the usual addition/subtraction/multiplication of functions. The additive identity element in this ring is the constant function 0, and multiplicative identity element is the constant function 1. (a) Let f: RR be a continuous function. Show that ƒ is a unit in C(R) if and only if the fucntion f has no zeroes. (b) Show that the ring C(R) contains zero divisors. That is, give an example of two non-zero continuous functions f, g: R→ R such that their product fg is the zero function.

4. Consider the set of all continuous functions f: R → R: 8(R) = {ƒ :R→R | ƒ is a continuous function}, This set forms a ring under the usual addition/subtraction/multiplication of functions. The additive identity element in this ring is the constant function 0, and multiplicative identity element is the constant function 1. (a) Let f: RR be a continuous function. Show that ƒ is a unit in C(R) if and only if the fucntion f has no zeroes. (b) Show that the ring C(R) contains zero divisors. That is, give an example of two non-zero continuous functions f, g: R→ R such that their product fg is the zero function.

Advanced Engineering Mathematics
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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4. Consider the set of all continuous functions f: R → R: 8(R) = {ƒ :R→R | ƒ is a continuous function}, This set forms a ring under the usual addition/subtraction/multiplication of functions. The additive identity element in this ring is the constant function 0, and multiplicative identity element is the constant function 1. (a) Let f: R → R be a continuous function. Show that ƒ is a unit in C(R) if and only if the fucntion f has no zeroes. (b) Show that the ring C(R) contains zero divisors. That is, give an example of two non-zero continuous functions f,g: R→ R such that their product fg is the zero function.
Transcribed Image Text:4. Consider the set of all continuous functions f: R → R: 8(R) = {ƒ :R→R | ƒ is a continuous function}, This set forms a ring under the usual addition/subtraction/multiplication of functions. The additive identity element in this ring is the constant function 0, and multiplicative identity element is the constant function 1. (a) Let f: R → R be a continuous function. Show that ƒ is a unit in C(R) if and only if the fucntion f has no zeroes. (b) Show that the ring C(R) contains zero divisors. That is, give an example of two non-zero continuous functions f,g: R→ R such that their product fg is the zero function.
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