Zero divisors: continuous functions over R

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Discussion Overview

The discussion centers on whether the ring of continuous functions over the real numbers contains zero divisors. Participants explore the implications of this question, seeking proofs or counterexamples related to the existence of nonzero continuous functions whose product is the zero function.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions if the ring of continuous functions over the real numbers has no zero divisors and seeks a proof or counterexample.
  • Another participant suggests that it should be possible to define two nonzero continuous functions whose product is the zero function.
  • A third participant expresses a desire to find simple examples of such functions and requests time to think about it.
  • A later reply notes that defining a piecewise function with continuous segments that match at endpoints can result in a new continuous function.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus; there are competing views regarding the existence of zero divisors in the ring of continuous functions.

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Does the ring of continuous functions over the real numbers have no zero divisors? If no 0 divisor, how can I prove it? Else, what is a counter example?
 
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you should be able to define two nonzero continuous functions whose product is zero function...
 
Wow, yea probably. I want to find easy ones. Give me until tomorrow. I'm dead tired. Thanks
 
Keep in mind if a function is defined piecewise such that each piece is continuous, and all the endpoints match up, you've defined a new continuous function
 

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