Let $$X$$ be a subset of $$\mathbb{R}$$, and let $$f: X \to \mathbb{R}$$ be a function. Let $$x_0$$ be an element of $$X$$. We say that $$f$$ is continuous at $$x_0$$ iff we have (in symbolic form):
in other words, the limit of $$f(x)$$ as $$x$$ converges to $$x_0$$ in X [...].
We say that $$f$$ is continuous on $$X$$ (or simply continuous) iff $$f$$ is continuous at $$x_0$$ for every $$x_0 \in X$$. We say that $$f$$ is discontinous at $$x_0$$ iff it is not continous at $$x_0$$.