Is this function x/sinx continuous?

[DrunkenPhD]

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

 

[jbunniii]

It is continuous wherever the denominator is nonzero, because it is the quotient of two continuous functions.

 

[DrunkenPhD]

Wolfram says NO :)
http://www.wolframalpha.com/input/?i=Is+f(x)=x/sinx+continuous+over+the+reals?

 

[jbunniii]

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.

 

[mathwonk]

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.

 

[WWGD]

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.

 

[AMenendez]

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.