## richyw 2 years ago show that f is continuous at every point, $$a\in\mathbb{R}$$

1. richyw

$f(x)=x\sin{\left(\frac{1}{x}\right)} \text{ if } x\neq0$$f(0)=0$

2. richyw

my textbook shows that$\lim_{x\rightarrow 0}x\sin{\left(\frac{1}{x}\right)=0}$because$\left|x\sin{\left(\frac{1}{x}\right)}\right|\leq|x|$

3. klimenkov

The function is continious in point $$x_0$$ if $$\lim \limits_{x\rightarrow x_0}f(x)=f(x_0)$$. You need to show that the function is continious in the point $$x=0$$.

4. richyw

I don't understand how this shows that the limit is zero?

5. richyw

I mean it's clear that this inequality holds to me. but it's unclear to me how this finds the limit.

6. klimenkov

Think about it. If $$\left|x\sin{\left(\frac{1}{x}\right)}\right|\leq|x|$$ and $$x\rightarrow0$$, so $$|x|$$ is very little, and then $$\left|x\sin{\left(\frac{1}{x}\right)}\right|$$ is little too!

7. richyw

ah, perfect. I get it now. so now because the limits exist, and f(0) is defined, and the limit= f(0), that should be satisfactory to show that the function is cts right?

8. klimenkov

Yes. That is right.

9. richyw

thank you so much

10. klimenkov

You are welcome.