question regarding continuity of sign function

Hi! I need to show that the sign function:

$\displaystyle f(x)=\left\{\begin{array}{lr}1:&x>0\\0:&x=0\\-1:&x<0\end{array}\right\}$

is discontinuous.

Is it true that $\displaystyle \lim_{x\to 0^-}f(x)=1$ and $\displaystyle \lim_{x\to 0^+}f(x)=-1$ so therefore $\displaystyle \lim_{x\to 0}f(x)$ does not exist? Im confused because doesnt the limit of f(x) at x=0 exist and is equal to 0?

Does that somehow mean that f(x) is continuous at 0 but discontinous at 0?