Is $\sqrt x \sin⁡\frac{1}{x}$ continuous at $0$? If it is not defined, does it count as continuous? Is $\sqrt x \sin⁡\frac{1}{x}$ continuous at $0$?
I found the limit of the function which is $0$, but the function is not defined at $0$. Is it continuous then?

If the function is undefined, it cannot be continuous. However, if the limit exists, you can define $g(x)$ to be $\sqrt{x} \sin(1/x)$ for $x \neq 0$ and let $g(0)=0$. Then $g$ would be continuous (provided you took the limit correctly).
