Showing continuity of a function by epsilon delta

Question

I have the function

$$f(x)= \begin{cases} x^2 \sin\left(\displaystyle\frac{1}{x}\right) & \text{if } x\ne 0 \\ 0 & \text{if }x=0 \end{cases} $$

and I have to show that it is continuous at $x_0=0$. I think they want me to use epsilon delta argument. Can somebody help?

Answer 1

Hint: use the squeeze method between $-x^2$ and $x^2$.

Answer 2

Hint:since $|\sin t|\le|t|$ for every t. See that $$|f(x)|\le|x|$$

Answer 3

I don't think they want $\delta-\epsilon$ proof. Im pretty sure that they want to proof the existence of $\lim_{x\to 0} f(x)$ and after show that $$\lim_{x\to 0} f(x)=0=f(0)$$ Aftter that you can conclude the continuity for the function!