Is this function x/sinx continuous?

1. Nov 19, 2014

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

2. Nov 19, 2014

jbunniii

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

3. Nov 19, 2014

DrunkenPhD

4. Nov 19, 2014

jbunniii

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.

5. Nov 19, 2014

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.

6. Nov 19, 2014

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.

7. Nov 21, 2014

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.

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