Continuity of $\sin(x)/x$

The function $f(x)=\frac{\sin x}{x}$ is not defined at $x=0$. However, its limit is $1$ as $x\to 0$, so we can define a continuous function:

$$g(x)=\cases{\frac{\sin x}{x} & $x\ne 0$\\ 1 & $x=0$}$$

Since the value of $g(0)$ is equal to $\displaystyle\lim_{x\to 0}g(x)$, then we have that $g$ is continuous at $0$.

It is undefined till you define it since the expression $\sin(x)/x$ is indeterminate at $x=0$. However it has a limit of $1$ as $x$ approaches zero. If you define a function to take the value $1$ when $x=0$ and the value $\sin(x)/x$ for $x\ne0$ then that function is continuous at zero.