Definition: A function \(f:\real^d \to \real\) is said to be continuous at a point \(\p\) if for all \(\eps > 0\), there exists \(\delta > 0\):
\[\norm{\x-\p} < \delta \implies \abs{f(\x) - f(\p)} < \eps.\]A direct consequence of this definition is that the limit of any continuous function is simply the function evaluated at its limit point:
Theorem: A function \(f:\real^d \to \real\) is continuous at a point \(\x\) if and only if \(\lim_{\x_n \to \x} f(\x_n) = f(\x).\)
Indeed, the above theorem is sometimes taken to be the definition of a continuous function.