Continuity of sin(1/x)

The continuity of this function is clear by the composition of the continuous functions. The question is " Can we extend this function by continuity on $0$?" The answer is NO. In fact the sequence $$x_n=\frac{1}{\frac\pi2+n\pi}$$ tends to $0$ but it's image by the given function is $(-1)^n$ hasn't a limit.

Your proof is correct; $\sin(1/x)$ is indeed continuous where it is defined.