I read this statement in "The Way of Analysis" -
*A function $f$ is continuous at a point $x_0$ in its domain that is a limit-point iff $d$ has a limit at $x_0$ and the value of $f(x_0)$ equals $\lim_{x \to x_0} f(x)$.
My question is - if the domain of the function has multiple limit points, does it mean that the function will not be continuous at the other limit points? Why not?
EDIT: I read that the limit of a function, if it exists, is unique. I understand that this means that the function can continuous at most one limit point. Is that not the case?