Let:

$\displaystyle f(x)=\left\{\begin{matrix} \mbox{sin}(\pi/x) & \mbox{} x \neq 0 \\ 0& \mbox{} x=0\end{matrix}\right$

Show that f is not continuous for x = 0

For both $\displaystyle \lim_{x \to 0^+} \ f(x) $ and $\displaystyle \lim_{x \to 0^-} \ f(x)$ we have that they are a number [-1,1]. We also see that $\displaystyle f(0)$ is not defined.

How to proceed?

What I know is that if $\displaystyle \lim_{x \to c^+} \ f(x) = \lim_{x \to c^-} \ f(x) = f(c)$, then the function is continuous. But what role does the $\displaystyle 0, x=0$ have?