Is this function x/sinx continuous?

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Discussion Overview

The discussion centers around the continuity of the function x/sinx, exploring its behavior across its domain and the implications of its definition at certain points. Participants examine whether the function can be considered continuous, particularly at points where the denominator is zero and at x=0.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions the continuity of the function x/sinx, noting a lack of examples in existing literature.
  • Another participant asserts that the function is continuous wherever the denominator is nonzero, as it is the quotient of two continuous functions.
  • Some participants reference Wolfram Alpha, which indicates that the function is not continuous over the reals, particularly at points where sinx equals zero (i.e., at x=nπ for any integer n).
  • A participant suggests that the function can be extended to be continuous at x=0, proposing that it could equal 1 at that point due to a finite limit.
  • There is agreement among participants that the main issue revolves around whether the function can be extended continuously to a real-valued function defined on the reals.
  • Another participant emphasizes that while the function is not continuous on the entire real line, it can be continuous when restricted to specific intervals.

Areas of Agreement / Disagreement

Participants express differing views on the continuity of the function x/sinx. While some agree that it is continuous wherever the denominator is nonzero, others highlight that it is not continuous on the entire real line and discuss the implications of extending the function at x=0.

Contextual Notes

The discussion includes limitations regarding the definition of continuity at points where the function is not defined, as well as the implications of extending the function at specific points.

DrunkenPhD
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Can we judge about continuity of function x/sinx??
Many examples in Google about sinx/x or xsinx but nothing about this function?
Is there any special case?
Regards
 
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It is continuous wherever the denominator is nonzero, because it is the quotient of two continuous functions.
 
DrunkenPhD said:
Wolfram says NO :)
http://www.wolframalpha.com/input/?i=Is f(x)=x/sinx continuous over the reals?
As I said, it is continuous wherever the denominator is nonzero. The denominator is zero at ##x=n\pi## where ##n## is any integer. The function is not even defined at these points, let alone continuous. It is defined and continuous everywhere else. Putting it another way, it is continuous on its domain.
 
it also has a slight extension which is continuous at x=0, since there is a finite limit there, namely extend the function to equal 1 at x=0.
 
I agree with previous posters; the issue is one of whether the function can be extended continuously into a Real-valued function defined on the Reals.
 
Exactly^^. The function as whole (i.e., on the domain (-∞, ∞)) is not continuous. However, if you restrict the domain and focus on specific intervals, then yes, it is continuous. Just look at a plot of the function for reassurance.
 

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