## Wallach Group Title I have the following function f(x)=sin(1/x) I've been told that the function is continuous for any x. I understand that the above function is continuous at (0,infinite) and also at (-infinite,0) but as much as I know the limit of this function while x -->0 does not exist or is undefined so how come the function is still continuous?? one year ago one year ago

1. terenzreignz Group Title

Perhaps it's for all x in its domain?

2. experimentX Group Title

the function is not continuous at x=0, are you missing something?

3. Wallach Group Title

it's for $f(x) \rightarrow R:R$

4. experimentX Group Title

if f(x) = x sin(1/x), the function could have been made continuous ... you might be interested in http://math.stackexchange.com/questions/199603/is-exp-left-1-over-x2-right-differentiable-at-x-0

5. Wallach Group Title

I might be missing some thing but I'm not sure what I would have uploaded the question but it's in Hebrew

6. Wallach Group Title

that I know. that's because x while x-->0 equals 0 and sin(1/x) is a Bounded function

7. Wallach Group Title

sry I think that the right way to translate the question is: does the Intermediate value theorem is true for sin(1/x) and if it does (and it does) explain

8. experimentX Group Title

bounded does not necessarily mean it is continuous. as x approaches to 0 the function oscillates violently between -1 and 1 without converging to any particular value. I don't think sin(1/x) is continuous at 0. Both because limit and value are not exactly defined.

9. Wallach Group Title

that's what I thought

10. experimentX Group Title

f(x) = x sin(1/x) is particularly interesting ... being continuous but not differentiable at x=0.

11. Wallach Group Title

but before I was talking about the following statement: when given the following limit $\lim_{x \rightarrow 0} f(x)*g(x) = ?$ if $\lim_{x \rightarrow 0} f(x) = 0$ AND g(x) is bounded then $\lim_{x \rightarrow 0} f(x)*g(x) = 0$

12. Wallach Group Title

which is the case that you're presenting

13. experimentX Group Title

yeah ... the case is different from what you are asking.

14. Wallach Group Title

but lets go back to this for a sec : does the Intermediate value theorem is true for sin(1/x) and if it does (and it does) explain because if think this is the correct translation to the question I was given

15. experimentX Group Title

the intermediate value theorem only applies for continuous function on some interval. Though for this particular Q i can say exactly that f(0) is between -1 and 1.

16. Wallach Group Title

right that's the thing the title of the question was is the Intermediate value theorem correct only for continuous function?? here are a few functions etc. etc.

17. Wallach Group Title

I think I understood the whole idea behind the question and I don't want to take too much of your time but any way thank you for your help :)

18. experimentX Group Title