show that f is continuous at every point, $a\in\mathbb{R}$

Question

richyw · Answer

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

richyw · Answer

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

klimenkov · Answer

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 $x=0$.

richyw · Answer

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

richyw · Answer

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

klimenkov · Answer

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!

richyw · Answer

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?

klimenkov · Answer

Yes. That is right.

richyw · Answer

thank you so much

klimenkov · Answer

You are welcome.