Discontinuity of $\sin\ (1/x) $ at $x=0$? Why is $\sin\ (1/x)$ discontinous at $x=0$ ? Isn't the value of sine always between $1$ and $0$? Why doesnot $\sin\ (1/x)$ exist?
The function $f(x)=\sin(1/x)$ isn't discontinuous at $x=0$, it is undefined. That's a different thing.

However, if you remedy that by defining it (say we set the value to $f(0)=0$), then it will necessarily be discontinuous. This follows quite immediately from (the negation of) any reasonable definition of continuity, for instance the $\epsilon$-$\delta$ definition using $\epsilon=\frac13$.
When x = 0, for sin(1/x) you have the fraction 1/0. Division by 0 is undefined, therefore there is a discontinuity at x= 0 + pik