If a 'function ' is continuous it must have its limit at $a$ equal to $f(a)$.
Considering tan(x) one may say that it is continuous for its domain but not a continuous function for all real numbers $\mathbb{R}$.
But isn't saying that wrong? Simply because if we take $\mathbb{R}$ as the input of our function our function is no longer a function because a function is defined when 'each' input value has a well defined output which is not the case for all inputs in $\mathbb{R}$. And thus our function is no more a function and we cannot say if it's a non-continuous 'function'
Best Answer
The domain of the tangent function is $\mathbb{R}\setminus\left(\frac\pi2+\pi\mathbb{Z}\right)$ and it is continuous. Since, if $k\in\mathbb Z$, the limit $\lim_{x\to\frac\pi2+k\pi}\tan x$ doesn't exist (in $\mathbb R$), you cannot extend it to a continuous function from $\mathbb R$ to $\mathbb R$.