## richyw Group Title show that f is continuous at every point, $$a\in\mathbb{R}$$ one year ago one year ago

1. richyw Group Title

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

2. richyw Group Title

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 Group Title

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 Group Title

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

5. richyw Group Title

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

6. klimenkov Group Title

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 Group Title

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 Group Title

Yes. That is right.

9. richyw Group Title

thank you so much

10. klimenkov Group Title

You are welcome.