I was wondering if you could help me with this: $ f(x)=\begin{cases} x^2\sin\left(\frac{1}{x}\right) & \text{ for } x \ne 0\\ 0 & \text{ for } x=0 \end{cases}. $ I need to observe that f is continuous on $\mathbb{R}$ and then explain why it is uniformly continuous bounded subset of $\mathbb{R}$.
Finally, is $f$ uniformly continuous on $\mathbb{R}$? Do I take $f'(x)$?
I know that I should be using the theorem for f being continuous, which says that f is continuous for some $x_0$ and then evaluating it for the entire $\text{dom}(f)$.
Apologizes for the format of my post!